Finding the Explicit Formula of a Geometric Sequence: Example 10

Topic

Sequences and Series

Description

Methodology for Finding the Explicit Formula

1. Identify the First Term: The first term of the sequence is denoted as a_1.
2. Determine the Common Ratio: The common ratio r is found by dividing the second term by the first term.
3. Write the Explicit Formula: The nth term of a geometric sequence can be found using the formula:

a_n = a₁•r^{(n - 1)}

       where a_n is the nth term, a₁ is the previous term, r is the common ratio, and n is the term number.

Given Sequence

Sequence: [5, 35, 245, 1715, 12005]

First term (a₁) = 5

Common ratio (r) = 35 / 5 = 7

Explicit formula: a_n = 5•7^{(n - 1)}

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