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Chapter 13: Problem 3

Write out \(S_{4}\) for each of the following, and determine whether it is true or false. $$S_{n}: \frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots+\frac{1}{2^{n}}=\frac{2^{n}-1}{2^{n}}$$

Short Answer

Expert verified
The statement is true for \(n = 4\).

Step by step solution

01

- Understand the sequence

The given sequence is a geometric series where each term is of the form \ \(\frac{1}{2^k}\) \ with the first term \(a = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\).
02

- Substitute n = 4

Write out the first four terms of the series: \ \(\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4}\).
03

- Calculate the left side of the equation

Evaluate each term and then sum them up: \ \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = 0.5 + 0.25 + 0.125 + 0.0625 = 0.9375\).
04

- Calculate the right side of the equation

Substitute \(n = 4\) into the given formula: \ \(\frac{2^4 - 1}{2^4} = \frac{16 - 1}{16} = \frac{15}{16} = 0.9375\).
05

- Compare both sides

Since both sides are equal (0.9375 = 0.9375), the given equation is true for \(n = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In our given problem, the sequence starts with \( \frac{1}{2} \) and continues with each term being multiplied by \( \frac{1}{2} \). The first few terms of the sequence are: \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), and \( \frac{1}{16} \). Each term (\

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Most popular questions from this chapter

Fill in each blank with the correct response. In each row of Pascal鈥檚 triangle, the first and last terms are _____, and each number in the interior of the triangle is the _____ of the two numbers just above it (one to the right and one to the left).

Find the indicated term for each geometric sequence $$ \frac{2}{3}, \frac{1}{3}, \frac{1}{6}, \ldots ; \quad a_{18} $$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$x^{2 n}+x^{2 n-1} y+\cdots+x y^{2 n-1}+y^{2 n}=\frac{x^{2 n+1}-y^{2 n+1}}{x-y}$$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$x^{2 n-1}+x^{2 n-2} y+\cdots+x y^{2 n-2}+y^{2 n-1}=\frac{x^{2 n}-y^{2 n}}{x-y}$$

Evaluate each expression. $$ { }_{13} C_{2} $$
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