Final answer:
The proof uses induction to show the validity of a geometric series equation. The base case is confirmed for n=1, and the induction step assumes the statement for n=k and proves it for n=k+1, completing the proof.
Step-by-step explanation:
To prove the geometric series 2+4+8+...+2^n equals 2^(n+1) - 2 using induction, we follow two main steps: the base case and the induction step.
Base case: Verify the statement for n = 1.
We get 2 = 2^(1+1) - 2, which simplifies to 2 = 4 - 2. This holds true, hence the base case is verified.
Induction step: Assume the statement is true for n = k, which means 2 + 4 + ... + 2^k = 2^(k+1) - 2. We need to prove it for n = k + 1.
The sum for k + 1 terms would be 2 + 4 + ... +2^k + 2^(k+1). According to our assumption, 2 + 4 + ... + 2^k = 2^(k+1) - 2, so we replace the sum of the first k terms with this expression to get 2^(k+1) - 2 + 2^(k+1).
Combining the terms, we have 2 * 2^(k+1) - 2, which simplifies to 2^(k+2) - 2. Hence, our statement holds for the (k + 1) term, completing the induction step.
By the principle of mathematical induction, the given series expansion equals 2^(n+1) - 2.