Problem 79 - 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 can be found by applying the formula \(S = \frac{a(1 - r^n)}{1 - r}\), where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Step 1: Understanding Geometric Sequences

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, 'r'. This can be represented as \(a, ar, ar^2, ar^3, \ldots, ar^{n-1}\) where 'a' is the first term and 'n' is the number of terms.

Step 2: Derive the Geometric Sum Formula

To find the sum of the first 'n' terms, we multiply the entire geometric sequence by 'r' and subtract the original sequence from it. Let \(S = a + ar + ar^2 + \ldots + ar^{n-1}\), then \(rS = ar + ar^2 + ar^3 + \ldots + ar^n\). Subtracting these equations, we get \((1-r)S = a - ar^n\). Therefore, the sum of the first 'n' terms, \(S\), of a geometric sequence can be found with the formula \(S = \frac{a(1 - r^n)}{1 - r}\) if \(r \neq 1\).

Step 3: Apply the Geometric Sum Formula

Given the first term 'a', the common ratio 'r', and the number of terms 'n', one can apply the formula for the sum of the first 'n' terms of a geometric sequence: \(S = \frac{a(1 - r^n)}{1 - r}\). This will provide the sum without needing to add all the terms together individually.