Problem 78 - College Algebra

Question

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

Step-by-Step Solution

Answer

The general term of a geometric sequence is given by the formula \(a * r^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio and \(n\) is the term number. To find any term, substitute the appropriate values into the formula.

Step 1: Recognize the Pattern

First, recognize the pattern followed in the geometric sequence. After the first term, each succeeding term is obtained by multiplying the previous term by the same non-zero number. This number is called the common ratio \(r\). The first term is denoted by \(a\).

Step 2: General Formulation

Formulate the general term of the sequence, denoted as \(nth\) term. In a geometric sequence, the \(nth\) term is calculated using the formula \( a * r^{n-1}\). Here, \(a\) represents the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term number.

Step 3: Application

Finally, to find any term in the sequence, substitute the values of \(a\), \(r\), and \(n\) in the formula \(a * r^{n-1}\)