Question

In Exercises $43-48,$ find the sum. $$\sum_{n=1}^{7} 2^{n}$$

   In Exercises $43-48,$ find the sum.
$$\sum_{n=1}^{7} 2^{n}$$

Contemporary Precalculus

Contemporary Precalculus

Thomas W.… 5th Edition

Chapter 12, Problem 43

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Solved by Expert Linh Vu

08/05/2020

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The sum of the first $n$ terms of a geometric series can be calculated using the formula: $$S_n = a \cdot \frac{1 - r^n}{1 - r}$$ Show more…

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In Exercises $43-48,$ find the sum. $$\sum_{n=1}^{7} 2^{n}$$

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Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the addition of a sequence of numbers defined by a specific rule or pattern. It employs the sigma symbol (?) along with defined limits to indicate the starting and ending indices of the terms, allowing for concise representation of long sums.

Geometric Series

A geometric series is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio. This type of series appears frequently in mathematics and its applications, and recognizing it enables the use of specialized formulas for summing its terms efficiently.

Formula for the Sum of a Finite Geometric Series

The finite geometric series sum formula provides a method to compute the sum of a geometric series without adding each individual term manually. The formula, S = a(r^n - 1)/(r - 1) for r ? 1, where a is the first term, r is the common ratio, and n is the number of terms, is pivotal in solving problems involving geometric progressions.

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