Problem 90 - College Algebra

Question

Explain how to find the general term of a geometric sequence.

Step-by-Step Solution

Answer

The general formula to find the n-th term in a geometric sequence is provided by \(a_n = a_1 \times r^{(n-1)}\) where \(r\) is the common ratio found by dividing a term of the sequence by its previous term. In our example, this gives the formula: \(a_n = 5 \times 3^{(n-1)}\)

Step 1: Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where any term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. Therefore, to find the general term of a geometric sequence, an understanding of the concept of ratio is pivotal. The formula to find the n-th term of a geometric sequence is \(a_n = a_1 \times r^{(n-1)}\) where \(a_n\) is the n-th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number

Step 2: Finding the ratio

The ratio \(r\) can be found by dividing any term in the sequence by the preceding term. For instance, if we have a sequence like 5, 15, 45, 135, ... We can find \(r\) by picking any term and dividing it by its previous term. Using the first two terms, \(r\) can be calculated as \(r = \frac{15}{5} = 3 \)

Step 3: Finding the general term

With the first term and common ratio at hand, we can now form a general formula for the terms of the sequence. From our previous example with \(a_1= 5\) and \(r = 3\), we substitute these values into the formula: \(a_n = a_1 \times r^{(n-1)}\), and so get the general term: \(a_n = 5 \times 3^{(n-1)}\)