Find each indicated sum. ∑i=14(−12)i\sum_{i=1}^{4} \left(-\frac{1}{2}\$$right)^{i}$$

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 35

9장, 문제 35

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

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Identify the sum notation: you need to find the sum of the terms from $ i = 1 $ to $ i = 4 $ of the expression $ \left(-\frac{1}{2}\right)^i $. This means you will add $ \left(-\frac{1}{2}\right)^1 + \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^3 + \left(-\frac{1}{2}\right)^4 $.

Recognize that this is a geometric series where the first term $ a = \left(-\frac{1}{2}\right)^1 = -\frac{1}{2} $ and the common ratio $ r = -\frac{1}{2} $.

Recall the formula for the sum of the first $ n $ terms of a geometric series: \[ S_n = a \cdot \frac{1 - r^n}{1 - r} \]. Here, $ n = 4 $.

Substitute the values into the formula: \[ S_4 = \left(-\frac{1}{2}\right) \cdot \frac{1 - \left(-\frac{1}{2}\right)^4}{1 - \left(-\frac{1}{2}\right)} \].

Simplify the numerator and denominator separately, then multiply by the first term to find the sum. Remember not to calculate the final numeric value, just set up the expression.

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주요 개념

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Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms. It specifies the index of summation, the starting and ending values, and the general term to be added. Understanding this notation is essential to correctly interpret and compute the sum.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Recognizing the series as geometric allows the use of formulas to find the sum efficiently, especially when the number of terms is finite.

Formula for the Sum of a Finite Geometric Series

The sum of the first n terms of a geometric series with initial term a and common ratio r (r ≠ 1) is given by S_n = a(1 - r^n) / (1 - r). This formula simplifies the calculation of sums without adding each term individually.

Find each indicated sum. ∑i=14(−12)i\(\sum\)_{i=1}^{4} \(\left\)(-\(\frac{1}{2}\]\right\))^{i}