Finding the Recursive Formula of an Arithmetic Sequence: Example 7

Topic

Sequences and Series

Description

Process for Finding the Recursive Formula

1. Identify the First Term: The first term of the sequence is denoted as a_1.
2. Determine the Common Difference: The common difference is found by subtracting the first term from the second term.
3. Write the Recursive Formula: The recursive formula for an arithmetic sequence is:

a_n = a_{n - 1} + d

       where a_n is the nth term, a_{n - 1} is the previous term, and d is the common difference.

Distinguishing Recursive from Explicit Formulas

1. Recursive Formula: Defines each term based on the previous term(s). It requires knowing the initial term and is useful for generating terms sequentially.
2. Explicit Formula: Allows direct computation of any term in the sequence without reference to previous terms. It is more efficient for finding terms far into the sequence.

Given Sequence

Sequence: [13, 21, 29, 37, 45]

First term (a₁) = 13

Common difference (d) = 21 - 13 = 8

Recursive formula: a_n = a_{n - 1} + 8

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