Question

What is the explicit formula for the arithmetic sequence in the table below?

Answer 1

This arithmetic sequence has a common difference of 4, meaning that we add 4 to a term in order to get the next term in the sequence.

The recursive formula for an arithmetic sequence is written in the form



For our particular sequence, since the common difference (d) is 4, we would write



So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence.

However, the recursive formula can become difficult to work with if we want to find the 50th term. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th. This sounds like a lot of work. There must be an easier way. And there is!

Rather than write a recursive formula, we can write an explicit formula. The explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use

an is the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations.

a1 is the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula.

n is treated like the variable in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be 12.

d is the common difference for the arithmetic sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for d into the formula. You must also simplify your formula as much as possible.