LEARN · EXPLAIN · REVISE

Prove the shifted-integer sum by induction.

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

\sum_{j=1}^n(j+5)=n(5)+n(n+1)/2

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)+k+1+t

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.
$n=1:1+5=5+1$
Apply the stated relation and retain its conditions.
$S_{k+1}=5k+k(k+1)/2+k+1+5$
Apply the stated relation and retain its conditions.
$S_{k+1}=5(k+1)+(k+1)(k+2)/2$
The next shifted term includes the fixed constant.

The requested relation or conclusion is shown below.

S_n=5n+n(n+1)/2

Checks and common pitfalls: The next shifted term includes the fixed constant.

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.