## Recognizing Patterns:

Can you find the particular pattern to each of the sequences below? See if you can guess the number that comes next for each set of numbers below!

a) 1, 1, 1, 1, 1, ?
b) 1, 2, 3, 4, 5, ?
c) 2, 4, 6, 8, 10, ?
d) 5, 10, 15, 20, ?

Easy to recognize what comes next, right? But how did you know? What pattern did each example make? That’s what we’ll be talking about in this post, sequences!

## What is a Sequence?

Let’s start with a formal definition. A sequence is a list of numbers or objects that are in a particular order and form a pattern. Each example above is a different type of sequence that follows a specific pattern! That’s why we were able to easily find the successive term of each sequence (solution to each example below).

The above examples of sequences are just a snippet of what a sequence can look like, but there are so many sequences that exist! There is even a website that compiles every sequence in the world possible, called The On-Line Encyclopedia of Integer Sequences (OEIS), it’s like a kind of dictionary but for sequences! They also accept new sequences to their website. After reading this post, see if you can come up with your own sequence to add to their encyclopedia!

## Types of Sequences:

There are so many different types of sequences! Sequences can take so many different forms, they can be infinite sequence where they can go on forever, or they can be finite sequence where they have an end. A sequence can be based on addition, subtraction, multiplication, division, or even based on the value of the previous term! Let’s take a look at each type of sequence one step at a time with an example for each:

## Infinite Sequences

Infinite Geometric Series

## Special Sequences

Fibonacci Sequence
Summing Every number from 1 to 100

## Arithmetic Sequence:

Arithmetic sequences are a sequence of numbers that form a pattern when the same number is either added or subtracted to each successive term. Take a look at the example of the of the arithmetic sequence below. Notice we are adding 2 to each term in the sequence 4, 6, 8, 10, … below. The number we add to each term in this sequence (in this case 2), is called the common difference. If we were to find the next term in the example of the sequence below, the next term of the sequence would be 12 (10+2=12).

Sometimes, we are asked to find not just the next term, but a term further down the line in the sequence. For example, what if we were asked to find the value of the 123rd term? That’s where the arithmetic sequence formula would come in! Check out the explicit formula below:

an=a1+(n-1)d

a1=First Term
n=Term Number in Sequence
d=Common Difference (Number Added/Subtracted to each Term in Sequence)

Now, to find the 123rd term of the above sequence, we would plug in the following values into our formula to solve. Notice the common difference here is still 2, because it is the number we are adding to all the terms in our sequence.

Notice the above arithmetic sequence is a an example of an infinite sequence! This means that we can continue finding terms for this sequence forever!

Want another example? For more examples and a step-by-step video on arithmetic sequences, check out the resources here and below:

## Geometric Sequence:

Geometric sequences are a sequence of numbers that form a pattern when the same number is either multiplied or divided to each subsequent term. Take a look at the example of a geometric sequence 4, 8, 16, 32, … below. Notice we are multiplying 2 to each term in the sequence below. The number that is multiplied to each term in the sequence is called the common ratio, which in the case of the geometric sequence below is 2. If the pattern were to continue, the next term of the sequence above would be 64, since 32 x 2=64.

Sometimes, we are asked to find not just the next term, but a term far down the line in the sequence, for example, what if we were asked to find the value of the 15th term? That’s where the geometric sequence formula would come in!

an=a1r(n-1)

a1 = First Term
r=Common Ratio (Number Multiplied/Divided to each Term in Sequence)
n= Term Number in Sequence

To find the 15th term of the above sequence, we would need to plug in the following values into our formula to solve:

Notice the above geometric sequence is a type of infinite sequence! This means that we can continue finding terms for this sequence, forever!

Want another example? For more examples and a step-by-step video on geometric sequences, check out the resources here and below:

## Recursive Sequences:

A Recursive Formula is a type of formula that forms a sequence based on the previous term value. These can be based on arithmetic sequences or a geometric sequences, the sequence type does not really matter, as long as each successive term is based on the previous term. What does that mean? Check out the example below for a clearer picture:

What are the first 5 terms of a recursive sequence given the following recursive formula:

First, let’s decode what this says:

Now, let’s see our formula in action! Notice below, we start with the first term, a1=2. Then we use 2 to plug into our formula for the second term to get 2+4=6, and we continue the pattern all the way through for the first five terms, to get our answer.

Recursive Sequence: Now that have used our recursive formula, notice it gave us this nice, beautiful sequence represented by each term circled in pink above, 2, 6, 10, 14, 18. That is our recursive sequence, and the answer to our question! Notice in this example, we have an infinite sequence that we can keep finding terms for, but since our question asked for the first 5 terms, we stopped here.

Want another example? For more examples and a step-by-step video on recursive sequences, check out the resources here and below:

## Finite Sequence and Series:

Now that we have gone over infinite sequences (Arithmetic, Geometric, Recursive), let’s dive into finite sequence and series!

What’s the difference between a sequence and a series? We know what sequences are, but what is this new word “series” all about? Well, what if we had the first 5 terms of a sequence, but now, we want to add them all together? That is a series, more specifically a finite series since we only want to add the first five terms! Another term you might see that describes a finite sequence is “Partial Sum” or “Partial Sums” because we are summing only part of the sequence.

## Finite Arithmetic Series:

A finite arithmetic series happens when we take the terms of an arithmetic sequence and we sum a finite number of them together. Basically, we know that the arithmetic sequence gives us the following terms:

But now we want to sum all these terms together:

How would we be able to find the partial sum of the first 20 terms of an arithmetic sequence? Well, we could sit there and crunch our numbers one by one on a calculator, or we could plug them into the arithmetic finite sequence formula below:

Sn=(n/2)(2a1+(n-1)d)

n=Number of Terms we want to Sum

d=Common Difference

a1=First Term

To find the sum of the first 20 terms of the above sequence, we would need to plug in the following values into our formula to solve. Notice that in this case, the common difference is 2, since each subsequent term is being added by 2.

For more examples and a step-by-step video on how to find the finite length of a Finite Arithmetic Series, check out the resources here and below:

## Finite Geometric Series:

A finite geometric series happens when we take the terms of a geometric sequence, and we sum them together. Basically, we know that the geometric series gives us the following terms:

But now we want to sum all these terms together:

How would we be able to find the partial sums of the first 20 terms of a geometric sequence? Well, we could sit there and crunch our numbers one by one on a calculator, or we could plug them into the formula below:

Sn=a1 (1-rn)/(1-r)

a1=First Term

n=Number of Terms we are Summing together

r=Common Ratio

For more examples and a step-by-step video on finite sequences and Geometric Series, check out the resources here and below:

## Infinite Sequences:

An infinite sequence is one in which the sequence just keeps going and going infinitely with no end. We have already seen examples of this earlier in this post when looking at an arithmetic sequence or geometric sequence. But now, we want to ask ourselves, what would happen if we were to add an entire infinite sequence, by adding together each term within the sequence?

When it comes to adding an infinite arithmetic sequence together, the arithmetic sequence always diverges to infinity. On the other hand, an infinite sequence that is also a geometric sequence, can either diverge to infinity, or converge to a number. If the idea of converging or diverging to infinity, doesn’t make sense yet, that’s ok, keep reading and we’ll go over everything!

## Infinite Geometric Series:

An infinite geometric series happens when we take the terms of a geometric sequence, and we sum them together, all of them together, starting with the first term, going all the way to infinity. Basically, we know that the geometric series gives us the following terms:

But now we want to sum all these terms together, from the first term all the way to infinity.

But wait, how do we know if it is even possible to sum together a geometric sequence starting with the first term and adding until infinity? Well, it all depends on divergence vs. convergence. If the common ratio is between -1 and 1, that means that the sum to infinity will converge and we can find the value of the sum of an infinite geometric sequence. If the common ratio has any other value otherwise it will diverge to infinity.

*Also here is a reminder of what a geometric sequence looks like, for anyone who needs it!

Now, let’s see if the following geometric series converges or diverges:

Since, we know it converges based on the common ratio, we know we can find the value of the infinite sum! Let’s use the formula below:

Sn=a1/(1-r)

a1=First Term

r=Common ratio

Next, let’s plug in our value into the formula:

For more examples and a step-by-step video on Infinite Geometric Series, check out the resources here and below:

## Fibonacci Sequence – The Most Famous Sequence!

The above sequence, known as the “Fibonacci Sequence,” is the most famous sequence in the world! How can a sequence be so famous? Why is it so special? Well, we’ll get to that in a minute, but before we do, can you see how the Fibonacci Sequence forms a pattern? What would be the next term?

Take your time, trying to figure out the next term of the sequence before taking a peak at the answer below:

## Why!? Solution Explained

The next term of the Fibonacci sequence is 21! The pattern of this famous sequence is all about adding the two previous terms together. That’s how we get 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, which brings us to get our missing term, 8+13=21. Take a look at how this sequence works below:

## Why is the Fibonacci Sequence Famous?

Now that we know the secret pattern behind this sequence, let’s look at why the Fibonacci sequence is so special! The Fibonacci sequence’s main claim to fame is that it is found throughout art, architecture and even in nature via the golden ratio.

The Golden Ratio is a proportion that is considered to be the most pleasing ratio to the human eye! You may also know the Golden Ratio as the golden mean, the divine proportion, phi, or the Greek letter ϕ. It is an infinite and irrational number that approximates to 1.618 and is found by adding two numbers together and then dividing by the larger number and if these same two numbers are then set equal to the larger number divided by the smaller number successfully, then the two numbers are a golden ratio equal to 1.618! If this sounds too confusing to imagine, just take a look at the formula below:

## Is it a Golden Ratio?

If the following formula holds true, then yes!

What is amazing about this ratio, is that it can be related back to the Fibonacci Sequence!

## The Golden Ration + Fibonacci Sequence:

If we were to take the sequential numbers found within the Fibonacci sequence (1,1,2,3,5..), and plug them into the golden ratio formula above, it would approximate to the golden ratio value, 1.618, the further along in the sequence we go.

## The Golden Ratio + Art + Architecture + Nature:

If we were to draw a rectangle that has golden ratio proportions, we would get the golden rectangle below.

Let’s draw a golden rectangle, within our golden rectangle to see what happens:

What happens if we continue this specific pattern and keep drawing in golden rectangles within itself?

Until eventually we get something like this….

The proportion between the width and height of these rectangles is 1.618 and can also be shown as the proportion between any two numbers in the Fibonacci sequence as the sequence approaches infinity.

The above pattern, when all lines are connected form a spiral that can be found within art, architecture, and nature itself! Below we have a picture of the Parthenon, the Mona Lisa, a shell, and the Taj Mahal.

## The Golden Ratio + YOU:

Want to know if you yourself has a face that fits the golden ratio?! Try measuring your face horizontally and vertically and plug in those values into the golden ratio formula, dividing the larger number by the smaller number. What did you get? Something close to 1.6 maybe!?

## The Golden Ratio + Resources:

If you’re looking to learn more about the Golden Ratio, be sure to check out these resources here.

## Donald Duck’s – Mathmagic Land:

Video shows how the golden ratio works and where we can find it! https://www.youtube.com/watch?v=U_ZHsk0-eF0&t=1s

## Investopedia + Golden Ratio:

Learn how the Fibonacci Sequence connects to the modern-day stock market! https://www.investopedia.com/articles/technical/04/033104.asp

## The Golden Ratio + Mathematics:

Learn where you will see the Golden Ratio in mathematic equations and shapes!
https://www.mathsisfun.com/numbers/golden-ratio.html

## Summing Numbers from 1 to 100:

Now that we’ve discussed the most famous sequence, let’s talk about the most well-known sequence, which are just plain old-fashioned natural numbers, starting at 1 and counting to infinity! The sequence we all grew up with and learned to love, yes, natural numbers:

1,2,3,4,5, ….100, …..

What if we were tasked with summing all the numbers from 1 to 100? How would we do that? Well, believe or not, there is a formula to adding all of these numbers together, that was discovered by a famous mathematician named Johann Karl Friedrich Gauss. Apparently, he came up with this pattern after being assigned to add all the numbers from 1 to 100 as an elementary school student! The teacher thinking this would be perfect “busy” work for her class did not see it coming!

## How to Sum 1 to 100:

Gauss found a pattern, when adding together the numbers 1 to 100. He looked at the big picture, recognizing that the sum between pairs of numbers from the beginning of the sequence and at the end of the sequence were the same throughout the entire list of numbers! For example, 1+100=101, 2+99=101, 3+98=101, and this pattern continues between all the numbers from 1 to 100! Take a look:

## Formula and Solution:

Noticing this pattern, lead Gauss to come up with the following formula to sum every number from 1 to 100:

Above we use the formula to sum every number from 1 to 100 but it can be used to find other sums as well! If you’re wondering where in the world the above formula even came from, try testing it out with a smaller sum, summing numbers from 1 to 10 and see what happens!

Just like we find the sum in an arithmetic or geometric series, we can find the sum of other types of sequences, as we see here with natural numbers! We can even realize that summing numbers 1 to 100 is like summing an arithmetic sequence with a common difference of 1. Hidden patterns and formulas are used to solve all different types of sequences we haven’t even gone over in this post. But let this post guide you with the basics! See if you can find the pattern of any sequences that you find (in this post, in class, or in the street walking around, as sequences are found there too).

If you’re looking for more resources, check out this website that connects Gauss’s Summation formula to arithmetic sequences found in the real world here.

I hope you find this overview of sequences helpful! If there is anything you’d like me to go over more or if you have any questions, please let me know in the comments. Also, please check us out on social media for the latest MathSux videos, lessons, practice, questions, cheat sheets, and more! Happy calculating!

## How to Make a Paper Cube Using Origami

Greeting math peeps and welcome to MathSux! In today’s post, we are going to have a bit of fun when finding the volume of a cube. We’ll start by going over how to make a paper cube using origami, then we will measure the dimensions of our real-world cube and find its volume. Hope you’re all feeling crafty and ready to take on this project with the Japanese art style of origami! If you don’t have any origami paper, please fill free to print out the origami guideline sheets included at the end of this post to follow along! Also, if the below written tutorial isn’t your style, watch the video up on YouTube and shown below. Hope you’re all having a great week and can find some fun in this post! Happy calculating!

## How to Make a Paper Cube Using Origami:

Step 1: Begin with six sheets of square origami paper (or cut out a square from any type of regular computer paper). If you want to really follow along, this paper is 4 inches by 4 inches and I have included a print out below.

Step 2: Fold the top of the origami paper in half, then unfold.

Step 3: Fold one side of the paper halfway to the center.

Step 4: Fold the left side to the center of the paper and rotate the paper horizontally.

Step 5: Take the bottom right corner and fold it to the top center point of the rectangle.

Step 6: Now fold the top left corner and bring it to the bottom edge, so both folds align in the middle.

Step 7: Next, Open both folds towards the center, undoing our last two steps.

Step 8: Open the first flap on top, and fold in the top right triangle.

Step 9: Unfold the bottom flap, and fold in the bottom left triangle.

Step 10: Now, we are going to fold down the top left corner to the center of the paper (just above the bottom flap).

Step 11: Fold up the bottom flap.

Step 12: Now take the bottom right corner, and tuck it in under the top flap, towards the center.

Step 13: Flip your piece of paper around.

Step 14: Fold the corner, towards the center.

Step 15: Now, fold in the bottom left edge, towards the center, to the top edge.

Step 16: Unfold, the left and right edges we made to get a shape like we have above.

Step 17: Repeat steps 1 through 16 5 more times! To have a total of six of the above shapes. We will need all of them to piece together our origami cube.

Step 18: Connect each of our shapes, by placing the edge of each in top left the corner “pocket” of each shape.

Step 19: Once they are all connected, we finally have created our origami cube!

Want to follow along with everything!? Check out the printable origami paper below to create your very own box!

## Volume of a Cube:

Now that we have made our cube using paper folding, we can measure one side of our cube and find its volume! Please note that these measurements are based on the 4″X 4″ origami paper I used. If you would like to follow along with your own project, just download and print the activity sheet above!

Did you create your origami cube with different-sized origami paper? Let us know what measurement you got for volume in the comments below!

Looking for more Origami + Math? Check out this post here to fold and find the volume of a real-world pyramid and rectangular prism. And for another math + crafty post, learn how to make a Mobius Band here!

What is your favorite way to combine math and the real world? Or math and art in general? Let me know in the comments and happy calculating!

Looking for another math challenge? Check out this post on derivatives here!

## 3 Equations 3 Unknown

Hi everyone and welcome back to MathSux! This week we are going to explore how to solve 3 equations 3 unknown. This is like the simultaneous equations we learned to love in Algebra, but now, on steroids. Previously, in algebra we only had to solve for two unknown variables and were given two equations. But now, in Algebra 2, we are faced with three simultaneous equations, as well as, three unknown variables (usually x,y, and z) that we must find the values of. These types of problems can look scary, but with the method of elimination and some practice, they are not so bad! Check out the video, step by step tutorial, and practice problems below to master this topic. Good luck and happy calculating!

## What are 3 Systems of Equations with 3 Unknown Variables?

Three systems of equations happen when there are three equations (usually with three unknown variables) are graphed or shown algebraically.  The graph can be represented by using the variables x,y,z (one for each of the three missing variables).  We are all used to the 2-dimensional coordinate plane with X going across and Y going up and down.  But now, with a new unknown variable, z, this gives us a new 3-dimensional axis. Instead of graphing lines like we are used to, these equations are going to be graphed as 3-dimensional planes.

As for finding the solution to a system of three equations, it is the same for any simultaneous equation, as the answer lies where all three planes intersect.  Below is an example of what a three-dimensional plane can look like when graphed. Don’t worry, typically you wont be asked to graph or even interpret anything like the picture below.

Instead of graphing, typically we are asked to find the solution using algebra! There are three equations and three unknown variables. How are we supposed to know and find the value for each variable?  Well, there are more than one way to solve these types of problems, but in this blog post and in the video above, we will be going over the elimination method.  This is the fastest and easiest way to solve for 3 unknown variables by hand.

## Solving by Elimination

The main idea of Elimination is to pick two pairs of equations and add them together with the goal of canceling out the same variable. We do this by lining up both equations one on top of the other and adding them together.  If a variable or variables does not easily cancel out at first,  we then multiply one of the equations by a number so it will cancel out.

For example, adding the first equation (2x+2y+z=20) and second equation (-3x-y-z=-18) will quickly cancel out z. When we pick another pair of equations to add, picking the second equation (-3x-y-z=-18) and the third equation (x+y+2z=16), we will also have to goal of canceling out z. Once z has been eliminated twice, it provides us with an opportunity to add together the two new equations we just found and eliminate even further to find the value of each unknown variable.  If this sounds like a confusing mouthful, do not worry! We’ll go over this process step by step below.

Check out how it’s done step by step below!

Step 1: First, let’s take another look at our equations and number them so we can keep track of which equations we’re using and when.

Step 2: Now let’s pick two equations and try to eliminate one of the variables.  If we look at the first two equations (labeled 1 and 2) above, notice we can easily cancel out at the unknown variable z.

Step 3: Ok great, what now?  Now we are going to take a different pair of our original equations and have the same goal of canceling out the same unknown variable z (just like we did in in the previous step).  Let’s pick the last two equations from above, (labeled 2 and 3). Notice that these two equations do not easily cancel out unknown variable z, therefore, we must multiply 2 times the whole of equation 2 so that they do cancel each other out.

Step 4: Now, that we’ve canceled unknown variable z from two different pairs of equations, we can use the two new equations we found in steps 2 and 3 (hi-lighted in purple) to cancel out another variable.

Step 5: Now that we have the value of one variable, x=3, we can plug it into one of our equations that we used in step 4.

Step 6: Last but not least, let’s find the value to our last unknown variable, z, by plugging in x=3 and y=5 into one of our original equations at the beginning of this example.

Was the wording a bit much? Too many things!? Don’t worry, check out the video and see how this questions gets worked through step by step on a piece of paper. And, if you get the example above, please check out the practice questions below!

## Practice Questions:

Find the values of each three variables for each system of equations:

## Solutions:

Want more MathSux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

*Bonus* Are 3 equations 3 unknown variables at the same time too much to handle? Check out how to find two simultaneous equations here!

## How to Construct Altitudes of a Triangle ⊿

Hi everyone and welcome back to another week of MathSux! For this week’s math lesson, I bring to you four constructions in one in How to Construct Altitudes of a Triangle using a compass and straight edge. In the video and post below we will define what an altitude is, find the altitude of an acute triangle, an obtuse triangle, and a right triangle. Just a reminder that an acute angle is an angle that is less than 90 degrees, an obtuse angle is an angle that is greater than 90 degrees and a right angle is an angle equivalent to 90 degrees. And lastly, within this post, we will use our construction to define and discover the orthocenter of a triangle, which is the point at which the altitude of each part of the triangle intersects.

I hope you find the below video helpful and interesting! Please let me know in the comments below if you have any questions. Good luck and happy calculating!

## What is an Altitude of a Triangle?

An Altitude is a perpendicular line drawn from the vertex of a triangle to the opposite side, creating a 90º angle (or right angle). Check out the example in the picture below, where the white dotted line drops down from the vertex of the triangle to the opposite side and how it forms a 90-degree angle represented by the white square. This is an altitude!

In the video above, we will look at how to find the altitude of an acute triangle, an obtuse triangle, and a right triangle step by step. Please watch the video above before reading the next part of this post about the orthocenter. This will makes things a bit clearer!

## How to Find the Orthocenter of Triangle with a Compass:

Now that we have found the altitudes of an acute triangle, obtuse triangle, and right triangle (in the video above), we can easily use our tools and knowledge of constructions and altitudes to find the orthocenter of a triangle.

The Orthocenter is a point where all three altitudes meet within a triangle. We can see in the example below, each dotted line represents an altitude to each vertex of the triangle.

In order to find the orthocenter using a compass, all we need to do is find the altitude of each vertex. The point at which they meet in the middle is known as the orthocenter. Check out the video above to see how this works step by step using a compass and straight edge or ruler.

## Constructions and Related Posts:

Looking to construct more than just the altitude of a triangle? Check out these related posts and step-by-step tutorials on geometry constructions below!

Construct an Equilateral Triangle

Perpendicular Line Segment through a Point

Angle Bisector

Construct a 45º angle

Square Inscribed in a Circle Construction

How to Construct a Parallel Line

Bisect a Line Segment

Construct a Parallel Line

## Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only \$7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

Still got questions? No problem! Don’t hesitate to comment with any questions below.

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Similar Triangles

Legs of a Right Triangle (when an altitude is drawn)

## Derivatives Practice Questions

Hi everyone and welcome to MathSux! In this weeks post, we will venture into Calculus for the first time! I won’t get too much into the nitty gritty explanation of what derivative are here, but instead will provide a nice overview of Derivatives Practice Questions. This post includes everything you need to know about finding the derivatives of a function including the Power Rule, Product Rule, Quotient Rule, and the Chain Rule. Below you will see examples, a Derivative Rules Cheat Sheet, and of course practice questions! I hope these quick examples help in the classroom or for that test coming up! Let me know if it helps and you want more Calculus lessons. Happy Calculating!

## What is a Derivative?

We use the derivative to find the rate of change of a function with respect to a variable. You can find out more about what a derivative is and its proper notation here at mathisfun.com. Read on below for a derivative rules cheat sheet, examples, and practice problems!

## Power Rule:

The power rule is used for finding the derivative of functions that contain variables with real exponents.  Note that the derivative of any lone constant number is zero.

## Product Rule:

The product rule is used to find the derivative of two functions that are being multiplied together.

## Quotient Rule:

Applying the quotient rule, will find the derivative of any two functions set up as a ratio.  Be sure to notice any numbers or variables in the denominator that can be brought to the numerator (if that’s the case, can use the more friendly power rule).

## Chain Rule:

The chain rule allows us to find the derivative of nested functions. This is great for trigonometric functions and entire functions that are raised to an exponent.

Ready for some practice questions!? Check out the ones below to test your knowledge of derivatives!

## Derivatives Practice Questions:

Find the derivatives for each function below.

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Also, if you find you need some motivation, check out my 6 tips and tricks for studying math here! Happy calculating! 🙂

*Also, if you want to check out Rate of Change basics click this link here!

## NYS Regents Review – Algebra June 2021

Greetings math friends! Today we are going to break down the NYS Regents, specifically the Algebra NYS Common Core Regents from June 2021, one question at a time. The following video playlist goes over each and every question one step at a time. I’ve been working on this playlist slowly adding new questions and videos every week and now that it is complete, it is time to celebrate (and/or study)! Please enjoy this review along with the study aids and related links that will also help you ace the NYS Regents. Happy calculating!

## Study Resources:

Looking to ace your upcoming NYS Regents!? Don’t forget to check out the resources below, including an Algebra Cheat Sheet, and important topics and videos to review. Good luck and happy calculating!😅

Algebra Cheat Sheet

Combing Like Terms and the Distributive Property

How to Graph an Equation of a Line

Piecewise Functions

NYS Regents – Algebra 2020

## How to Study Math?

How is one supposed to study math!? Well, there is usually only one way, and that is to practice, practice, practice! But don’t get too stressed, because you can also make practice fun (or at least more pleasant).

Add some background music to your study session and make a nice cup of tea before diving in for the brain marathon. Another idea is to study only 1 hour at a time and to be sure to take breaks. Can’t seem to get that one question? Take a break and walk a way, or even better find a new study spot! It’s been scientifically proven that studying in different places can boost your memory of the very information you’re trying to understand.

What study habits do you have that have worked for you in the past? Let me know in the comments and good luck on your upcoming test!

** Be sure to check out the full list of Algebra lessons and old Regents questions review!**

## Legs of a Right Triangle (when an altitude is drawn)

Greetings math peeps and welcome to another week of MathSux! In todays post we are going to explore how to find the legs of a right triangle when an altitude is drawn from the vertex to its hypotenuse. We are going to take this step by step on how to solve a problem like this. Although, I go over the long way to solve this problem, there is also short cut many people use called the “geometric means” which is also briefly mentioned in this post (under the Tip! section). Whichever method you choose, do what makes most sense to you! Happy calculating!

## Similar Right Triangles (with Altitude drawn):

When two triangles have equal angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but if they have the same angles and the sides are in proportion, they are similar! Triangles can be proven similar by AA, SAS, or SSS. For more on similar triangles, check out this post here.

There is a special type of scenario that happens with similar right triangles. When an altitude is drawn from the vertex of a right triangle, it forms two smaller triangles, which creates three right triangles that are similar to the original triangle, based on Angle Angle (AA). Check out the example to see how it works!

In triangle ABC, an altitude is drawn from angle A to its hypotenuse BC. Notice that this creates three 90º right triangles in total (ABC, ABD, and ADC).

Drawing an altitude created three total right triangles, broken out below:

If you take a closer look, at the triangles above you’ll notice that these two new triangles (ABD and ADC) share an angle with the original bigger triangle ABC and a 90º angle. This makes each of the new triangles similar to the original triangle by AA. We can therefore say that:

## How are the New Right Triangles Similar?

Ready for an Example?  Lets check out the one below!

Step 1: To find the length of the missing legs of a right triangle, first, let’s separate each right triangle to see what we’re working with, along with the values of each length that was provided.

Step 2: We want to find the length of side AC, so let’s use the triangles whose sides include AC.  This leads us to use triangles ABC and ADC.

Step 3:  Now, let’s set up our proportion to find our missing side.  To make it easier for ourselves let’s first flip and rotate triangle ADC to line up with triangle ABC.

Think you’re ready to try some practice questions on your own? Check out the ones below!

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating!

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Similar Triangles

How to Construct the Altitude of a Triangle with Compass

## Infinite Geometric Series Formula

Hey math friends and happy Wednesday! Today we are going to take a look at the infinite geometric series formula in summation notation. First, we’ll review what a geometric sequence (or geometric progression) is, then look at the difference between divergence and convergence based on the common ratio, and then finally look at the formula. The unique thing about infinite geometric sequences is that the formula doe snot always apply. We only know if the formula applies to our series based on the value of the common ratio.

This may sound a bit complicated, with a few new concepts, but fear not as divergence and convergence are easy to identify! If you have any questions, don’t forget to check out the video and practice problems below. Happy calculating! 🙂

## What does it mean to find the “Sum of Infinite Geometric Sequence” (Series)?

We already know what a geometric sequence is: a sequence of numbers that forms a pattern when the same number is multiplied or divided by each successive term.

The number is multiplied or divided to each term is called the common ratio. We can see below that each number in the sequence is being multiplied by 1/2. Therefore, we can say that the common ratio of the geometric sequence below is 1/2.

Example:

But then, what happens if we wanted to sum all the terms of our geometric sequence together? The sequence of terms is now considered to be an infinite geometric series since we are summing all of the terms of our series together, infinitely. Take a look below:

Example:

## Summation Notation:

We can also write our infinite geometric series using something called Summation Notation shown below. This is a different format, also known as sigma notation, if you need a refresher on how to use and read summation notation, please check out the link in this paragraph. When writing in summation notation, we are actually writing the sum of the geometric series in a different way. Check it out below:

Now that we know what each part of this summation notation looks like, let’s actually identify what each part of this equation means in relation to our geometric sequence:

a1=The first term. In this case, we can see that the first term of our geometric sequence is the number 5 in the example below. Therefore, we can say a1=5.

r= The common ratio is the number that is multiplied or divided by each consecutive term within the sequence. In the example below, each number is multiplied by 1/2, therefore we can say, r=1/2.

n= The total number of terms we are trying to sum together. In the example mentioned below, we are trying to sum an infinite number of terms, so in this case, n starts at 1 and then continues infinitely to increase by 1 for each subsequent term.

If we were to expand our inifite series summation it would look something like this:

But how would we calculate this? This is an infinite series represented in summation notation, therefore, it goes on infinitely, how does this work?

That’s where our Infinite Geometric Series formula comes in handy! But before we jump into finding a solution and looking at our formula, let us take a closer look at what types of geometric series qualify for using the infinite series formula in the first place.

## Divergence Vs. Convergence: When do we use the Infinite Series formula?

There are two types of Infinite Geometric Series:

Type 1: Infinite Geometric Series that diverges to infinity

Type 2: Infinite Geometric Series that converges to a numeric value (-1 < r < 1)

Check out the differences in the example below:

## Infinite Geometric Series Formula:

Now the moment you’ve all been waiting for… the Inifite Geometric Series Formula! Remember, the use of this formula is purely based on the common ratio of our geometric sequence.

The infinite geometric series formula can only be used if the common ratio, r, is less than 1 and greater than -1. If the common ratio, r, is not between -1 and 1, then the sum of the geometric sequence diverges to infinity (and the formula cannot be used).

Now that we have the formula to work with and know when to use it (when -1 < r < 1), let’s take another look at our question and apply our infinite geometric series formula to find a solution:

Step 1: First, let’s identify the common ratio to make sure that it is between numbers -1 and 1. If the common ratio is a fraction between these two numbers, then we can use our infinite series formula.

Step 2: Now that we know we can use our infinite series formula, let’s write out each part of the formula and identify what numbers we are going to plug in.

Step 3: Last but not least, let’s fill in our formula with the correct numbers and solve with the given values.

Think you are ready to try inifite geometric series examples on your own? Check out the following infinite geometric series practice questions below to determine if a series converges or diverges.

## Practice Questions:

State if each given series converges or diverges, then if applicable find the solution.

## Related Posts:

Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the finite series posts next including: finite geometric sequence or finite arithmetic series!

Arithmetic Sequence (arithmetic progression)

Recursive Formula

Finite Arithmetic Series

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

Still, got questions? No problem! Don’t hesitate to comment below or reach out via email. And if you would like to see more MathSux content, please help support us by following ad subscribing to one of our platforms. Thanks so much for stopping by and happy calculating!

## Back to School Review

Hey math friends and welcome back to MathSux! Back to school season is upon us which means most students (and teachers) will need to review a bit before diving into a completely new subject.  In order to alleviate some of the back to school whoas, I bring to you, this back to school review! Check out the videos below to get the math juices flowing whether you’re new to Algebra, Geometry, or Algebra 2/Trig! I hope you find these videos helpful and wish everyone the best of luck in their first days at school! Happy calculating! 🙂

## How to Prepare for Algebra:

Calling all incoming algebra students, Combining Like Terms is a great place to start! You most likely have combined like terms before, but there’s nothing like sharpening your skills before getting the intense Algebra questions that are coming your way. Check out the video below and try the practice questions here!

## How to Prepare for Geometry:

Geometry students, you have the world of shapes ahead of you! It’s an exciting time to review basic Area, Perimeter, Circumference, and Pythagorean Theorem rules before moving ahead with this subject. Review the Pythagorean Theorem below from Khan Academy and check out the last page of the review sheet here to review area and perimeter.

## How to Prepare for Algebra 2:

Relieve the fond memories of algebra by reviewing all the different ways to Factor and Solve Quadratic Equations! This is a great way to prepare for rational expressions and the harder algebra 2 problems that are right around the corner.  Check out the video below and related practice questions here to reinforce these hopefully not yet forgotten algebra skills!

Hope you find this quick review helpful before diving in for the real deal! Besides brushing up on these math topics, what type of new school year routines do like to practice in your classroom or at home? Let me know in the comments and happy calculating! 🙂

## Olympics Statistics: Top 10 Medals by Country

Greetings and welcome back to MathSux! This week, in honor of the Tokyo Olympics, I will be breaking down some Olympic Statistics. We will look at the top 10 countries that hold the most medals and then look at the top 10 medals earned by country in relation to each country’s total population. Let’s take a look and see what we find! Also, please note that all data used for this analysis was found on the website, here. Anyone else watching the Olympics? Try downloading the data with the link above and see what type of conclusions you can find! Happy Calculating! 🙂

## Top 10 Countries: Total Olympic Medals

Below shows the top 10 total medals earned by country from the beginning of the Olympics in 1896 to present day July 2021. As we can see in the graph below, the United States is way ahead of the game with thousands more Olympic medals when compared to any other country in the entire world! I always knew the U.S. did well in the Olympics, but did not realize it was to this magnitude!

## Top 10 Countries: Total Olympic Medals Based on Population

Below is a different kind of graph. This percentage rate represents total medals earned over time from 1896 to July 2021 divided by the country’s total population. In this case, we can see that Lichtenstein has earned way more medals based on their small population size when compared to any other country in the world! This is amazing and unexpected!

Remember that all data for the above graphs were made from the following website, here. Are you surprised by the above graphs and conclusions? Try downloading the data on your own and see what you can conclude using your own Olympics Statistics skills! Happy calculating! 🙂

Looking to apply more math to the real world? Check out how to find volume of the Hudson Yards Vessal in NYC here