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Chapter 14: Problem 24

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

Short Answer

Expert verified
The sum of the given geometric sequence is \(-\frac{37}{81}\)

Step by step solution

01

Identify the First Term

Since in our summation, we start from \(i = 2\), our first term, denoted as \(a\), would be \(\left(-\frac{1}{3}\right)^2=-\frac{1}{9}\)
02

Identify the Common Ratio

The common ratio, denoted as \(r\), is the base of the exponent in our sum, which is \(-\frac{1}{3}\)
03

Identify the Number of Terms

The number of terms, denoted as \(n\), is the difference between the upper and lower limit of the sum plus one (since both are inclusive). Let's denote the upper limit as \(N\) and the lower limit as \(M\), then \(n = N - M + 1 = 4 - 2 + 1 = 3\)
04

Calculate the Sum of the Geometric Sequence

The sum \(S\) of a finite geometric sequence can be calculated using the following formula: \[S = a \times \frac{1 - r^n}{1 - r}\] Substituting the values \[S = -\frac{1}{9} \times \frac{1 - (-\frac{1}{3})^3}{1 - (-\frac{1}{3})}= -\frac{37}{81}\]

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Most popular questions from this chapter

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