## What Is An Arithmetic Sequence?

An arithmetic sequence is an infinite sequence of numbers in which the difference between each pair of consecutive numbers is always the same. For example, in the sequence 1, 3, 5, 7, 9 . . . the difference between one number and the next is always 2.

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What is the constant difference (*d*) between any two consecutive numbers in the following sequences?

- -5, -3, -1, 1, 3, 5 . . .

- .5, 1, 1.5, 2 . . .

- 10, 6, 2, -2 . . .

In the first sequence, *d* = 2 because you can add 2 to any number in the sequence to get the next number. For example, -3 + 2 = -1 and 1 +2 = 3. In the second sequence, *d* = .5. In the third sequence, each number is 4 less than the previous number, so *d* = -4.

**The Recursive Formula For An Arithmetic Sequence**

One way of finding a number within a sequence is to use the recursive formula. To write the formula, we use the following notation:

*a* is a term in the sequence.

*n* is the number of terms in the sequence.

*d* is the constant difference between terms.

Thus, *a _{n}* =

*a*+

_{n-1}*d*

In other words, to find the 5th number in a sequence with a constant difference of 6, we need to know the 4th number (*a _{n-1}*) and add 6 to it. If we are given the sequence 5, 11, 17, 23, and we need to find the next number, we can easily apply this formula by adding 6 to 23 and getting 29. In other words, if

*d*= 6 and if

*a*– 1 = 23, then

_{n}*a _{n}* = 23 + 6 = 29

**The Explicit Formula For An Arithmetic Sequence**

If we only have the first number in a sequence (*a _{1}*), however, the explicit formula can be a more useful way to find another number in the sequence. To understand how the explicit formula is derived, let’s start with the following sequence where

*d*= -7:

100, 93, 86, 79 . . .

To get the first number, we start with 100 and add -7 zero times. So a_{1 }= 100 + (-7 x 0). To get the second number, we subtract 7 one time. So a_{2} = 100 + (-7 x 1). The next number in the series is a_{3} = 100 + (-7 x 2), and so on. Each time we are adding -7 exactly one less time than the number of terms in the sequence. Therefore, we can write a general formula to express this pattern as follows:

*a _{n}* =

*a*+ (

_{1}*n*-1) x

*d*

If we want to find, for example, the 17th number in a series that begins with 3 and has a constant difference of .5, we can plug that information into the formula like this:

a_{17} = 3 + (17-1) x .5 = 11

**Practice**

*Problem 1*: What is the constant difference (*d*) in the following sequence?

24, 32, 40, 48, 56 . . .

*Solution:* In this sequenced = 8 because we can add 8 to each number to get the next number.

*Problem 2:* What is the next number in the sequence above?

*Solution:* Using the recursive formula, we know that the 6th number (*a _{6}*) is equal to the 5th number (

*a*) plus the constant difference (

_{6-1}*d*). Since 56 + 8 = 64, the next number in the series is 64.

*Problem 3:* Write an explicit formula for the sequence in Problem 1 and use that formula to find the 11th number in the sequence.

*Solution:* Since *a _{n}* = 24 + (n-1) x 8, and n = 11, then

*a*= 24 + (11-1) x 8 = 104.

_{11}### Recursive formula:

A **recursive formula** is a formula that defines each term of a sequence using the preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.

Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence is the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.

Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a **recursive formula**, a formula that defines the terms of a sequence using previous terms.

A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining $a n $ in terms of preceding terms. For example, suppose we know the following:

We can find the subsequent terms of the sequence using the first term.

So the first four terms of the sequence are ${3, 5, 9, 17}$ .

The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.

To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so

### MUST THE FIRST TWO TERMS ALWAYS BE GIVEN IN A RECURSIVE FORMULA?

*No. The Fibonacci sequence defines each term using the two preceding terms, but many recursive formulas define each term using only one preceding term. These sequences need only the first term to be defined.*