In Exercises 11–24, 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. Step 1: Understand the statement to prove by induction. We want to prove that for every positive integer $n$, the sum $1\; +\; 2\; +\; 2^2\; +\; \backslash cdots\; +\; 2^\{n-1\}\; =\; 2^n\; -\; 1$ holds true.
2. Step 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 they are equal.
3. Step 3: Inductive Hypothesis - Assume the statement is true for some positive integer $k$, that is, assume $1\; +\; 2\; +\; 2^2\; +\; \backslash cdots\; +\; 2^\{k-1\}\; =\; 2^k\; -\; 1$.
4. Step 4: Inductive Step - Using the inductive hypothesis, prove the statement is true for $k\; +\; 1$. Start with the sum up to $k$ and add the next term $2^k$, then simplify to show it equals $2^\{k+1\}\; -\; 1$.
5. Step 5: Conclude that since the base case is true and the inductive step holds, by mathematical induction, the statement is true 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 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. The formula for the sum of the first n terms is S_n = (r^n - 1)/(r - 1), which simplifies here to 2^n - 1.

Exponents and Powers of Two

Understanding exponents is crucial, especially powers of two, which grow exponentially. The expression 2^(n-1) represents the (n-1)th power of 2. Recognizing how to manipulate and simplify expressions involving powers of two helps in verifying the equality and performing induction steps.