*Contributor: Ashley Nail. Lesson ID: 13636*

Discover how to identify, evaluate, and write explicit and recursive functions to solve for terms in a sequence.

categories

subject

Math

learning style

Auditory, Visual

personality style

Lion, Otter, Beaver, Golden Retriever

Grade Level

High School (9-12)

Lesson Type

Quick Query

- We know that construction workers and engineers need to study a lot of math, but do you know how they can use sequences and functions in their jobs?

Imagine they are building a multi-floor apartment building with a parking garage. The construction workers are only on the 3rd floor, but the engineers need to find out how tall the building will be after the 27th floor and the 28th floor.

- How can we use functions to help the engineers?

Let's look at an arithmetic sequence:

{2, 7, 12, 17, 22 …}

In this sequence of positive integers, we can see that 5 is added to each term to get the next term in the sequence.

- How can we write this relationship as a function?

Well, there are two ways: an **explicit sequence** or a **recursive sequence**.

First, let's look at an **explicit** **sequence**.

In order to follow along, let's name each term. We will use the letter *t*.

- The first term is t
_{1}is 2. - The second term is t
_{2}is 7. - The third term is t
_{3}is 12. - The fourth term is t
_{4}is 17. - The last and fifth term of this sequence is t
_{5}is 22.

Now we are ready to write an explicit function for this sequence.

We already know we are using *t* to represent a term in the sequence. Let's use *k* to represent which term in the sequence we are looking for.

To write an equation that will work for any term in the sequence, we are looking for what t_{k} equals:

t_{k} =

The first term in the sequence is 2, and we know we add 5 each time. So let's start with that:

t_{k} = 2 + 5

We can also tell that we are adding by 5 one less times the number term we are looking for.

In other words, imagine if we are looking for the fourth term in the sequence, we are adding 5, 3 times.

- Which is one less than 4 (fourth term or t
_{4})?

t_{k} = 2 + 5(k -1)

Then, we will rewrite this sequence in function notation. We will swap t_{k} with t(k) to show how we will input into this function to solve for an output value:

t(k) = 2 + 5(k -1)

Let's test it!

t(1) = 2 + 5(1 - 1)

t(1) = 2 + 5(0)

t(1) = 2 + 0

t(1) = 2

The first term of this sequence is 2. Our explicit function works!

Test for the second, third, and fourth terms also! (t_{2}, t_{3}, and t_{4})

You just learned about explicit sequences!

Now let's use the same sequence to write a **recursive** **sequence**. All this means is that the function builds off of the previous term in the sequence.

We know the first term in the sequence is 2 or t_{1} = 2

To build off of the first term, you will add 5 to one less than the number in the sequence:

t_{k} = t_{k-1} + 5

So now, let's solve for the second term in the sequence or t_{2}:

t_{2} = t_{(2-1)} + 5

t_{2} = t_{1} + 5

Plug in what we already know as t_{1}:

t_{2} = 2 + 5

t_{2} = 7

So, according to this recursive function, the second term in the sequence should be 7, and it is!

Now that we know t_{2} = 7, we can solve for t_{3} using the same recursive function.

Now, since this is an infinite sequence, use both the explicit and recursive function to find the 6th term in this sequence.

- Now if you wanted to find the 72nd term in this sequence, what function would be easiest to use?
- The explicit or the recursive? Why?

- What are some of the differences between these two types of functions?

With the explicit function, we can find any term in the sequence. With the recursive function, we have to go in order, or at least know the value of the previous term.

Now you are ready to practice identifying and interpreting these types of functions and sequences!

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