Arithmetic Sequence Recursive Formula

Before delving into the arithmetic sequence recursive formula, it's essential to understand the concept of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each successive term is obtained by adding a fixed number to the previous term. This fixed number is known as the common difference and is denoted by "d." For example, in the sequence -1, 1, 3, 5, ..., the common difference is 2, as each term is obtained by adding 2 to the previous term. Now, let's explore the arithmetic sequence recursive formula through examples.

What Is Arithmetic Sequence Recursive Formula?

Recursion in the context of an arithmetic sequence involves determining one of its terms by applying a consistent rule to its preceding term. As previously established, each term in an arithmetic sequence is derived by adding a constant value (referred to as the common difference, denoted as "d") to its preceding term. Consequently, the arithmetic sequence recursive formula is as follows:

Arithmetic Sequence Recursive Formula

Here,represents the nth term of the arithmetic sequence.

a n represents the nth term of the arithmetic sequence.

a n − 1 represents the (n - 1)th term of the arithmetic sequence, which is the term immediately preceding the nth term.

d = The common difference (the difference between every term and its previous term).

Also Check - Area Formulas

Examples Using Arithmetic Sequence Recursive Formula

Example 1:

Let's find the first four terms of the sequence when a1 = –4 and the recursive formula is given by an = an−1 + 5.

Starting with a1:

n = 1

a1 = -4

Now, we can use the recursive formula to find the subsequent terms:

n = 2

a2 = a2−1 + 5

a2 = -4 + 5

a2 = 1

n = 3

a3 = a3−1 + 5

a3 = 1 + 5

a3 = 6

n = 4

a4 = a4−1 + 5

a4 = 6 + 5

a4 = 11

So, the first four terms of the sequence are: -4, 1, 6, 11.

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Example 2:

Let's find the recursive formula for the sequence 2, 4, 6, 8, 10, ... .

In this sequence, we can observe that each term is 2 more than the previous term.

So, the recursive formula can be written as:

a1 = 2 (the first term)

an = an−1 + 2 (each term is 2 more than the term before)

This recursive formula represents the given sequence.