LEARN · EXPLAIN · REVISE

Prove the finite power-of-two identity.

Read the idea, work independently, then explain what changed.

高二選擇性必修第二册(A版).pdf · 4.4 · PDF 49 / printed page 44

TOPIC 01

Mathematical induction

Build proofs with a verified base, an explicit induction hypothesis and a valid step.

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

1+2+\cdots+2^{n-1}=2^n-1

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Verify the base case, assume the claim at k, and prove it at k+1.

Hint 2

Use this intermediate relation.

S_{k+1}=S_k+2^k

Worked solution

Verify the base case, assume the claim at k, and prove it at k+1.
Apply the stated relation and retain its conditions.
$n=1:\ 1=2-1$
Apply the stated relation and retain its conditions.
$S_{k+1}=2^k-1+2^k=2^{k+1}-1$
At n terms the highest exponent is n−1.

The requested relation or conclusion is shown below.

S_n=2^n-1

Checks and common pitfalls: At n terms the highest exponent is n−1.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Which condition is essential in mathematical induction?
Can a correct induction step rescue a false or missing base case?

Defining relation: Build proofs with a verified base, an explicit induction hypothesis and a valid step.
$P(n_0)\land[P(k)\Rightarrow P(k+1)]$
Conditions: Induction proves a statement only for the specified integer domain starting at the base case.