How to Find the Explicit Formula for an Arithmetic Sequence: Step-by-Step Guide

Explicit Formula for Arithmetic Sequence

To find the explicit formula for an arithmetic sequence, we need two pieces of information: the first term (a₁) and the common difference (d)

To find the explicit formula for an arithmetic sequence, we need two pieces of information: the first term (a₁) and the common difference (d).

The general form of an arithmetic sequence is:

aₙ = a₁ + (n – 1) × d

In this formula, aₙ represents the nth term of the sequence, a₁ is the first term, n is the position/index of the term we want to find, and d is the common difference between consecutive terms.

Let’s break down the formula and understand each part:

1. a₁: This is the first term of the sequence. It is given or can be determined from the problem.

2. (n – 1): This represents the position of the term we want to find, minus 1. For example, when n = 1, (n – 1) equals 0. When n = 2, (n – 1) equals 1. And so on.

3. d: This is the common difference between consecutive terms. It remains constant throughout the sequence. It is given or can be determined from the problem.

Now, let’s apply the formula to an example:

Example: Find the 7th term of the arithmetic sequence where the first term is 2 and the common difference is 3.

Using the formula:

aₙ = a₁ + (n – 1) × d

a₇ = 2 + (7 – 1) × 3
= 2 + 6 × 3
= 2 + 18
= 20

So, the 7th term of the arithmetic sequence is 20.

Note that you can use this formula to find any term in the arithmetic sequence by simply substituting the given values of a₁, d, and n into the formula.

Remember, an arithmetic sequence is a sequence of numbers where the difference between consecutive terms remains constant. The explicit formula helps us express the nth term using the first term and the common difference.

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