Howdy math peeps! In this post, we are going to go over the recursive formula step by step by reviewing an old Regent’s question. These things may look weird, confusing, and like a **“what am I doing?”** moment, but trust me they are not so bad! We are going to take a look at the Regents question below and then find the correct recursive formula that follows the given sequence by going through each answer choice. Before we begin to answer our questions though, we will first define and break down what a recursive sequence is. Also, be sure to check out the video at the end of this post for more examples and further explanation. Happy calculating!

Before we dive into the solution to this question, let’s first look at a recursive formula example and define what they are in the first place!

**What is the Recursive Formula?**

A **Recursive Formula** is a formula that forms a sequence based on the previous term value. All this means is that it uses a formula to form a sequence-based pattern. Recursive formulas can take the shape of different types of sequences, including arithmetic sequences (sequence based on adding/subtracting numbers) or geometric sequences (sequence based on multiplying/dividing numbers). Check out the example below:

a_{1}=2 , a_{n+1}=a_{n}+4

## How do You Solve a Recursive Equation?

When solving a recursive equation, we are always given the first term and a formula. We start by using the first given term of our sequence, usually represented as a_{1}, and plug it into the given formula to find the value of the second term of the sequence. Then we take the value of the previous term (the second term in our sequence) and plug it into our formula again, to find the third value of our sequence…. and the pattern continues! The solutions we get from each step up forms a sequence. If this sounds confusing, don’t worry because we are going to look at an example! Take a look at the recursive formula below:

a_{1}=2 , a_{n+1}=a_{n}+4

Now let’s take another look at our recursive formula, this time breaking down what each part means:

a_{1} always represents the first given term, which in this case is a_{1}=2. Next, we plug in 2 for a_{n} into our formula a_{n}+4 to get (2)+4=6. This gives us our second term in the sequence, which is 6. Next, we plug in 6 into the formula to get (6)+4=10, which is the value of the third term in the sequence. And we continue the pattern, always taking the value of the preceding term! In this case, I just found the first few terms below, but the recursive formula can continue infinitely! Again , if this sounds confusing, please take look at the pattern below:

**Terms of the Sequence:** 2, 6, 10, 14, 18….

Notice all the terms of the sequence are circled in pink, forming a sequence of, **2, 6, 10, 14, 18**!? This is what the recursive formula produces!

## Now back to our Original Question:

Q: What recursively-defined function represents the sequence 3, 7, 15, 31,

(1) f(1) = 3, f(n + 1) = 28 (n) +3

(2) f(1) = 3, f (m + 1) = 28(0) – 1

(3) f(1) = 3, f(n + 1) = 28 (n) +1

(4) f(1) = 3, f (n + 1) = 38 (n) -2

**How do I answer this question?**

At first glance, all of these answer choices may look exactly the same as there are many recursive formulas to choose from. The first thing we need to do is to identify how each answer choice is different. Notice below, the section highlighted in green? This is what we will focus on for finding the correct recursive formula!

The question we are working with actually gives us a sequence and we need to find the recursive formula that works with it! Our goal with this question is to work backward to test out each recursive formula given to us until we get the correct sequence. To begin, let’s first identify each term in our given sequence.

As we go through each answer choice, we are looking for the recursive formula that gives us the above sequence 3, 7, 15, 31. Let’s start with choice (1) which happens to be a type of geometric sequence.

Right away we can see the sequence forming for choice (1) is 3, 11, … where the first term, 3, matches our original sequence, but the second term we get which is, 11, does not. This means we will need to move on and find the sequence of the next option, choice (2), which happens to be another type of geometric sequence. Let’s take a look:

For choice (2), we can see that the sequence we get is 3, 7, 127 which matches our given sequence for the first two terms 3, 7, but does not match the third term 127, when we need a 15 here (the original sequence is 3,7,15,31). Thus, we must move onward to the next recursive formula by testing out choice (3), which this time is an arithmetic sequence! Maybe we will have better luck!

This last option, choice (3) provided us with the same sequence we were originally provided within the original question, 3, 7, 15, 31. We have found our answer and now we can celebrate!

## Recursive Formula Examples

Still have questions about recursive formulas? Check out more on recursive formula examples here and in the video above! And if you’re looking to learn all there is to know about sequences, check out this post here! Looking to move ahead? Check out the infinite geometric series lesson here! Also, please don’t hesitate to comment with any questions. Happy calculating! 🙂

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