Problem 36 - College Algebra

Question

Find each indicated sum. $$\sum_{i=2}^{4}\left(-\frac{1}{3}\right)^{i}$$

Step-by-Step Solution

Answer

The sum of the series is -\frac{37}{243}.

Step 1: Understand the series

In this problem, we are working with a geometric series whose formula is represented as \(-\frac{1}{3}^i\), where \(i\) is the step. This series begins at \(i = 2\) and continues until \(i = 4\)

Step 2: Calculate the terms

Calculate each term in the series individually. To do it, replace \(i\) in the formula with the each individual value from 2 to 4. For \(i=2\), the term is \(-\frac{1}{3}^2 = -\frac{1}{9}\). For \(i=3\), the term is \(-\frac{1}{3}^3 = -\frac{1}{27}\). For \(i=4\), the term is \(-\frac{1}{3}^4 = -\frac{1}{81}\).

Step 3: Sum up all the terms

Finally, sum up all the individual terms together to get the sum of the series. The sum therefore is \(-\frac{1}{9} - \frac{1}{27} - \frac{1}{81} = - \frac{37}{243}\).