In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 2 + 4 + 8 + ... + 2^n = 2^(n+1) - 2

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 left-hand side of the equation: 2. Check if it equals the right-hand side: 2^(1+1) - 2.
2. **Step 2: Inductive Hypothesis** - Assume the statement is true for some positive integer k. That is, assume 2 + 4 + 8 + ... + 2^k = 2^(k+1) - 2.
3. **Step 3: Inductive Step** - Prove the statement is true for n = k + 1. Start with the left-hand side: 2 + 4 + 8 + ... + 2^k + 2^(k+1).
4. Use the inductive hypothesis to replace the sum up to 2^k: (2^(k+1) - 2) + 2^(k+1).
5. Simplify the expression to show it equals the right-hand side for n = k + 1: 2^(k+2) - 2.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves 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 particularly useful for proving statements about integers.

Geometric Series

A geometric series is a series of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the context of the given question, the series 2 + 4 + 8 + ... + 2^n can be recognized as a geometric series with a common ratio of 2. Understanding the sum of geometric series is essential for simplifying and proving the statement.

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 given statement, the terms involve powers of 2, which are exponential in nature. Recognizing the properties of exponential functions, such as their growth rates and how they relate to sums, is crucial for manipulating and proving the equation.