Problem 51 - Beginning and Intermediate Algebra

Question

Find the sum of the first six terms of the geometric sequence with \(a_{1}=9\) and \(r=2\)

Step-by-Step Solution

Answer

The sum of the first six terms of the geometric sequence is \(S_6 = 567\).

Step 1: Identify the given information

We are given the following information: - First term (\(a_1\)) = 9 - Common ratio (r) = 2 - Number of terms (n) = 6

Step 2: Use the formula for the sum of a geometric sequence

The formula for the sum of the first n terms of a geometric sequence is: $$ S_n = \frac{a_1(r^n - 1)}{r-1} $$ Substitute the given values of \(a_1\), r and n into the formula and solve for the sum \(S_n\).

Step 3: Substitute the values into the formula and solve

Substitute the given values into the formula: $$ S_6 = \frac{9(2^6 - 1)}{2-1} $$ Now simplify and calculate the sum: $$ S_6 = \frac{9(64 - 1)}{1} $$ $$ S_6 = \frac{9(63)}{1} $$ $$ S_6 = 9(63) $$ $$ S_6 = 567 $$ The sum of the first six terms of the geometric sequence is 567.