Problem 91 - College Algebra

Question

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.

Step-by-Step Solution

Answer

The sum of the first \(n\) terms of a geometric sequence without adding all the terms can be found by using the formula \(S_n = a\frac{1-r^n}{1-r}\) where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 1: Understanding Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'common ratio'. It can be represented as \(a, ar, ar^2, ar^3, ..., ar^{n-1}\). Here, \(a\) is the first term and \(r\) is the common ratio.

Step 2: Sum formula of Geometric Sequence

The formula to find the sum, \(S_n\), of the first \(n\) terms of a geometric sequence is \(S_n = a\frac{1-r^n}{1-r}\) for \(r≠1\). This formula simplifies the process of adding up all the terms.

Step 3: Applying the Formula

Given a geometric sequence and the number of terms \(n\), plug the values into the formula to find the sum. Suppose the first term \(a = 1\), the common ratio \(r = 2\), and the number of terms \(n = 3\). Then, \(S_n = 1\frac{1-2^3}{1-2} = -7\).