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

Verified step by step guidance

1. Step 1: Understand the statement to prove by induction. We want to prove that for every positive integer $n$, the sum $2\; +\; 4\; +\; 8\; +\; \backslash cdots\; +\; 2^n$ equals $2^\{n+1\}\; -\; 2$.
2. Step 2: Base Case - Verify the statement for $n\; =\; 1$. Calculate the left side: $2^1\; =\; 2$. Calculate the right side: $2^\{1+1\}\; -\; 2\; =\; 2^2\; -\; 2\; =\; 4\; -\; 2\; =\; 2$. Since both sides are equal, the base case holds.
3. Step 3: Inductive Hypothesis - Assume the statement is true for some positive integer $k$, that is, $2\; +\; 4\; +\; 8\; +\; \backslash cdots\; +\; 2^k\; =\; 2^\{k+1\}\; -\; 2$.
4. Step 4: Inductive Step - Prove the statement for $k\; +\; 1$. Start with the left side for $k\; +\; 1$: $2\; +\; 4\; +\; 8\; +\; \backslash cdots\; +\; 2^k\; +\; 2^\{k+1\}$. Using the inductive hypothesis, replace the sum up to $2^k$ with $2^\{k+1\}\; -\; 2$, so the expression becomes $(2^\{k+1\}\; -\; 2)\; +\; 2^\{k+1\}$.
5. Step 5: Simplify the expression from Step 4: $(2^\{k+1\}\; -\; 2)\; +\; 2^\{k+1\}\; =\; 2^\{k+1\}\; +\; 2^\{k+1\}\; -\; 2\; =\; 2\; \backslash cdot\; 2^\{k+1\}\; -\; 2\; =\; 2^\{k+2\}\; -\; 2$. This matches the right side of the statement for $n\; =\; k\; +\; 1$, completing the inductive step and proving the statement for all positive integers $n$.

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 found by multiplying the previous term by a constant ratio. In this problem, the series 2 + 4 + 8 + ... + 2^n 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, grow exponentially and are fundamental in this problem. Recognizing how to manipulate and simplify expressions involving powers of two is essential for proving the given equality.