Problem 31 - College Algebra

Question

Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{8} 3^{i}$$

Step-by-Step Solution

Answer

The sum of the geometric sequence for the first 8 terms is 19683.

Step 1: Identifying the Parameters

In the geometric series \(\sum_{i=1}^{8} 3^{i}\), the first term \(a = 3^1 = 3\), the common ratio \(m = 3\), and the number of terms \(n = 8\).

Step 2: Using the Sum of Geometric Sequence Formula

We know that the formula for the sum of the first \(n\) terms of a geometric sequence is \(S_n = a(m^n - 1) / (m - 1)\) where \(a\) is the first term, \(m\) is the common ratio and \(n\) is the number of terms. Substituting the values found in Step 1 into the equation results in \(S_8 = 3(3^8 - 1) / (3 - 1)\).

Step 3: Calculating the Sum

Now, we just have a simple evaluation problem. After calculating, we see that \(S_8 = 3(3^8 - 1) / (3 - 1) = 19683\).