Problem 35 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{4}\left(-\frac{1}{2}\right)^{i}$$

Step-by-Step Solution

Answer

The sum of the geometric series is \( -\frac{15}{32} \).

Step 1: Identify necessary elements

In the given series \( \sum_{i=1}^{4}\left(-\frac{1}{2}\right)^{i} \), the first term \( a \), can be found by substituting \( i = 1 \) i.e., \( a = (-\frac{1}{2})^{1} = -\frac{1}{2} \). The common ratio \( r = -\frac{1}{2} \) and the number of terms \( n = 4 \).

Step 2: Use the geometric sequence sum formula

Substitute \( a = -\frac{1}{2} \), \( r = -\frac{1}{2} \) and \( n = 4 \) into the geometric sequence sum formula \( S = a * \frac{1 - r^n}{1 - r} \).

Step 3: Calculate the sum

Input values into the equation gives us: \( S = -\frac{1}{2} \times \frac{1 - (-\frac{1}{2})^{4}}{1 - -\frac{1}{2}} \). Solving gives: \( S = -\frac{1}{2} \times \frac{1 - \frac{1}{16}}{1.5} = -\frac{15}{32} \)