Arithmetic Sequences: Learn It 3

Using Explicit Formulas for Arithmetic Sequences

Now that you’ve written the terms of the sequence, your next task is to write the formula that represents the arithmetic sequence.

explicit formula for an arithmetic sequence

An explicit formula for the [latex]n\text{th}[/latex] term of an arithmetic sequence is given by

[latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex]

Try to use the pattern you’ve identified in the terms to create a formula that can generate any term in the sequence.

How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, [latex]{a}_{2}-{a}_{1}[/latex].
  2. Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
Write an explicit formula ([latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex]) for the arithmetic sequence:

[latex]\left\{2\text{, }12\text{, }22\text{, }32\text{, }42\text{, }\ldots \right\}[/latex]

Using Recursive Formulas for Arithmetic Sequences

Now that you’ve written the explicit formula, let’s explore a different approach: the recursive formula. Try writing a formula that defines each term of the sequence based on the previous term. Remember, the recursive formula builds the sequence step by step, starting from the first term.

recursive formula for an arithmetic sequence

The recursive formula for an arithmetic sequence with common difference [latex]d[/latex] is:

[latex]\begin{align}&{a}_{n}={a}_{n - 1}+d && n\ge 2 \end{align}[/latex]

How To: Given an arithmetic sequence, write its recursive formula.

  1. Subtract any term from the subsequent term to find the common difference.
  2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.
Write a recursive formula for the arithmetic sequence.

[latex]\left\{-18,-7,4,15,26, \ldots \right\}[/latex]