In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1

Verified step by step guidance

1. <Step 1: Base Case> Verify the statement for the initial value, usually n=1. Substitute n=1 into the equation: 1 = 2^1 - 1. Check if both sides are equal.
2. <Step 2: Inductive Hypothesis> Assume the statement is true for some positive integer k, i.e., 1 + 2 + 2^2 + ... + 2^(k-1) = 2^k - 1.
3. <Step 3: Inductive Step> Prove the statement for n=k+1 using the inductive hypothesis. Start with the left side of the equation for n=k+1: 1 + 2 + 2^2 + ... + 2^(k-1) + 2^k.
4. <Step 4: Simplify Using Hypothesis> Replace the sum 1 + 2 + 2^2 + ... + 2^(k-1) with 2^k - 1 (from the inductive hypothesis). The expression becomes (2^k - 1) + 2^k.
5. <Step 5: Simplify Further> Combine like terms: (2^k - 1) + 2^k = 2 * 2^k - 1 = 2^(k+1) - 1. This matches the right side of the equation for n=k+1, completing the induction.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements, typically involving positive integers. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is essential for proving formulas or properties that are defined recursively.

Geometric Series

The expression 1 + 2 + 2^2 + ... + 2^(n-1) represents a geometric series where each term is a power of 2. The sum of a geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. In this case, the first term is 1 and the common ratio is 2, which simplifies the calculation of the series sum.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In the context of the given equation, 2^n - 1 represents an exponential function where the base is 2. Understanding the properties of exponential growth is crucial for analyzing the behavior of the series and proving the equality through induction.