Problem 32 - College Algebra

Question

Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{6} 4^{i}$$

Step-by-Step Solution

Answer

The sum of the first 6 terms of the sequence is 5460.

Step 1: Identifying the Parameters

First identify the first term \(a\), the common ratio \(r\), and the number of terms \(n\). Here, \(a = 4^1 = 4\), \(r = 4\) (since each term is multiplied by 4 to get the next one) and \(n = 6\).

Step 2: Substitute in the Formula

Next, substitute the values of \(a\), \(r\), and \(n\) into the formula for the sum of the first \(n\) terms of a geometric sequence. You get \(\frac{4(4^6 - 1)}{4-1}\).

Step 3: Simplify the Terms

Simplify the fraction by doing the calculations. The numerator becomes \(4 \times 4095 = 16380\), and the denominator is \(3\).

Step 4: Complete the Division

Finally, perform the division to get the sum of the sequence. The result is \(5460\).