Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 17

Chapter 9, Problem 17

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2² + ... + 2^n-1 = 2^{n - 1}

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Identify the statement to prove using mathematical induction: For every positive integer $n$, the sum $1 + 2 + 2^{2} + \dots + 2^{n-1} = 2^{n} - 1$ holds true. Note that the problem's right side should be $2^{n} - 1$ instead of $2^{n-1}$ to match the sum of a geometric series.

Base Case: Verify the statement for $n=1$. Substitute $n=1$ into the left side and right side of the equation and check if both sides are equal.

Inductive Hypothesis: Assume the statement is true for some positive integer $k$, that is, assume $1 + 2 + 2^{2} + \dots + 2^{k-1} = 2^{k} - 1$.

Inductive Step: Using the inductive hypothesis, prove the statement for $n = k + 1$. Start with the sum up to $k+1$ terms: $1 + 2 + 2^{2} + \dots + 2^{k-1} + 2^{k}$. Replace the sum up to $k$ terms with the inductive hypothesis and simplify the expression.

Show that the simplified expression equals $2^{k+1} - 1$, which completes the inductive step and proves the statement for all positive integers $n$ by mathematical induction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. In this problem, the series 1 + 2 + 2^2 + ... + 2^(n-1) is geometric with ratio 2. Understanding the formula for the sum of a geometric series helps in verifying the closed-form expression.

Exponents and Powers of Two

Exponents represent repeated multiplication of a base number. Powers of two, like 2^(n-1), grow exponentially and are common in sequences and series. Recognizing how to manipulate and simplify expressions involving powers of two is essential for proving the given equality.

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Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 22 + ... + 2n-1 = 2n - 1

Verified video answer for a similar problem:

Key Concepts

Mathematical Induction

Geometric Series

Exponents and Powers of Two

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2² + ... + 2^n-1 = 2^{n - 1}