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Chapter 10: Problem 29

Induction is not the only method of proving that a statement is true. Exercises \(26-29\) suggest alternate methods for proving statements. Prove that \(1+4+4^{2}+\cdots+4^{n-1}=\frac{4^{n}-1}{3}\) by using the formula for the sum of terms of a geometric sequence.

Short Answer

Expert verified
The equation \(1+4+4^{2}+\cdots+4^{n-1}=\frac{4^{n}-1}{3}\) is proved by using the sum of geometric sequence formula, \(S_n = a \frac{(r^n-1)}{r-1}\).

Step by step solution

01

Identify Parameters of the Geometric Sequence

The first term \(a = 1\) and the common ratio \(r = 4\). The number of terms \(n\) is specified as an arbitrary integer.
02

Apply the Sum of Geometric Sequence Formula

Substitute \(a = 1\), \(r = 4\), and \(n\) into the formula \(S_n = a \frac{(r^n-1)}{r-1}\). The equation becomes \(S_n = 1 \frac{(4^n - 1)}{4 - 1}\)
03

Simplify the Equation

On simplifying the equation obtained from Step 2 gives you the desired result \(S_n = \frac{4^n - 1}{3}\). Hence, the equation \(1+4+4^{2}+\cdots+4^{n-1}=\frac{4^{n}-1}{3}\) is proved.

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Key Concepts

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

Proof by Induction
Proof by induction is a powerful mathematical technique used to prove the validity of a proposal for all natural numbers. It consists of two main steps. First, the base case is proven, where the statement is shown to be true for the initial value, often for the number 1. Then, the inductive step is shown, where the statement assumed to be true for a number k is used to prove the statement for k+1. This technique is akin to a chain reaction; once the initial domino falls (the base case), the following ones (the inductive step) continue indefinitely, thus proving the statement for all natural numbers.
Geometric Progression
A geometric progression (GP), also known as a geometric sequence, is a series 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. For example, the sequence 2, 6, 18, 54, ... is a geometric progression where each term is three times the previous term. The behavior of a GP is largely dictated by the common ratio. If the ratio is greater than one, the terms will grow exponentially, but if it's between zero and one, the terms will diminish towards zero. This characteristic leads to diverse applications, from modeling exponential growth/decay to the analysis of loan amortization.
Sum of a Geometric Series
The sum of a geometric series is the total value obtained by adding all the terms of a geometric progression. For a geometric series with first term a, common ratio r (which is not equal to 1), and n number of terms, the sum is given by the formula \( S_n = a \frac{r^n - 1}{r - 1} \). When the ratio r is greater than 1, the series diverges as n approaches infinity; however, if r is between -1 and 1, the series converges to a finite value, defined as \( S = \frac{a}{1 - r} \). Understanding this formula is essential as it appears in various fields, such as finance for calculating compound interest, or physics for analyzing waves.
Mathematical Proof
A mathematical proof is a logical argument demonstrating the truth of a mathematical statement. Proofs use a series of agreed-upon premises and deductive reasoning to arrive at a conclusion, ensuring the statement's validity without any doubt. It's the foundational mechanism by which mathematical propositions are ascertained to be universally true. Beyond induction, proofs can be constructed through direct demonstration, contradiction, contrapositive, and construction, among other techniques. The art and science of proving statements are what confer mathematics its rigor and certainty.

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Most popular questions from this chapter

State whether the sequence is arithmetic or geometric. $$4,10,16,22, \dots$$

A slot machine has four reels, with 10 symbols on each reel. Assume that there is exactly one cherry symbol on each reel. Use this information and the counting principles from Section 10.4. What is the probability of getting exactly three cherries?

In this set of exercises, you will use sequences to study real-world problems. Investment An income-producing investment valued at \(\$ 2000\) pays interest at an annual rate of \(6 \% .\) Assume that the interest is taken out as income and therefore is not compounded. (a) Make a table in which you list the initial investment along with the total value of the investment-related assets (initial investment plus total interest earned) at the end of each of the first 4 years. (b) What is the total value of the investment-related assets after \(n\) years?

Recreation The following table gives the amount of money, in billions of dollars, spent on recreation in the United States from 1999 to \(2002 .\) (Source: Bureau of Economic Analysis) $$\begin{aligned} &\text { Year } \quad 1999 \quad 2000 \quad 2001 \quad 2002\\\ &\begin{array}{l} \text { Amount } \\ \text { (S billions) } 546.1 \quad 585.7 \quad 603.4 \quad 633.9 \end{array} \end{aligned}$$ Assume that this sequence of expenditures approximates an arithmetic sequence. (a) If \(n\) represents the number of years since 1999 , use the linear regression capabilities of your graphing calculator to find a function of the form \(f(n)=a_{0}+n d, n=0,1,2,3, \ldots,\) that models these expenditures. (b) Use your model to project the amount spent on recreation in 2007

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