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 is \( \frac{7}{243} \).

Step 1: Substitute 'i' with 2

Substituting 'i' with 2, we get \(-\frac{1}{3}^{2}\), which equals to \( \frac{1}{9} \)

Step 2: Substitute 'i' with 3

When 'i' is 3, we get \(-\frac{1}{3}^{3}\), which equals to \(-\frac{1}{27} \)

Step 3: Substitute 'i' with 4

Substituting 'i' with 4 results in \(-\frac{1}{3}^{4}\), which equals to \( \frac{1}{81} \)

Step 4: Sum up the results

Adding all calculated values for \(i = 2\), \(i = 3\), and \(i = 4\) we get the total sum: \( \frac{1}{9} - \frac{1}{27} + \frac{1}{81}\).

Step 5: Simplify the sum

The sum simplifies (using common denominator) to \( \frac{9 - 3 + 1 }{243} = \frac{7}{243} \).