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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주요 개념

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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.

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