LEARN · EXPLAIN · REVISE

Prove n³−n+3tn is divisible by 3 for all n≥1.

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

TOPIC 01

Sequences: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

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

01 / Foundation#Your turn

t=23

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.

f(k+1)−f(k)=3k(k+1)+3t

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.
$f(1)=3(23)$
Apply the stated relation and retain its conditions.
$f(k+1)=f(k)+3k(k+1)+3(23)$
Both the base and the increment are multiples of three.

The requested relation or conclusion is shown below.

3 | (n^3−n+3tn)

Checks and common pitfalls: Both the base and the increment are multiples of three.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment