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

Verified step by step guidance

1. 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.
2. 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.
3. Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(1 + 2 + 2^{2} + \dots + 2^{k-1} = 2^{k} - 1\).
4. 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.
5. 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.

Key Concepts

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.