LEARN · EXPLAIN · REVISE

Prove the scaled telescoping identity by induction.

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 / Standard#Your turn

\sum_{j=1}^n31/[j(j+1)]=31n/(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)+t/[(k+1)(k+2)]

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.
$n=1:31/2=31/2$
Apply the stated relation and retain its conditions.
$S_{k+1}=31k/(k+1)+31/[(k+1)(k+2)]=31(k+1)/(k+2)$
The constant factor remains in the added term and target.

The requested relation or conclusion is shown below.

S_n=31n/(n+1)

Checks and common pitfalls: The constant factor remains in the added term and target.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment