a_{n}=a_{1}r^{(n-1)}

**a _{1}** = First Term

**r**=Common Ratio (Number Multiplied/Divided to each Term in Sequence)

**n**= Term Number in Sequence

Hi everyone and welcome to Mathsux! In this post, we’re going to go over **geometric sequences**. We’ll see what geometric sequences are, break down their formula a_{n}=a_{1}r^{(n-1)}, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

**What are Geometric Sequences?**

**Geometric sequences **are a sequence of numbers that form a pattern when the same number is either *multiplied* or *divided* to each subsequent term.

Example:

Notice we are multiplying 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 64. This is a geometric sequence!

In this sequence it’s easy to see what the next term is, but what if we wanted to know the 15^{th} term? That’s where the Geometric Sequence formula comes in!

**Geometric Sequence Formula:**

Now that we broke down our geometric sequence formula, let’s try to answer our original question below:

->First, let’s write out the formula:

-> Now let’s fill in our formula and solve with the given values.

Let’s look at another example where, the common ratio is a bit different, and we are dividing the same number from each subsequent term:

-> First let’s identify the common ratio between each number in the sequence. Notice each term in the sequence is being divided by 2 (or multiplied by 1/2 ).

-> Now let’s write out our formula:

-> Next let’s fill in our formula and solve with the given values.

**Practice Questions:**

- Find the 12
^{th}term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, …. - Find the 17
^{th}term given the following sequence: 3, 9, 27, 81, 243,….. - Find the 10
^{th}term given the following sequence: 5000, 1250, 312.5, 78.125 ….. - Shirley has $100 that she deposits in the bank. She continues to deposit twice the amount of money every month. How much money will she deposit in the twelfth month at the end of the year?

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out **arithmetic sequences** click this link here! And if you want to learn about even more sequences, check out the link here!