LEARN · EXPLAIN · REVISE

Prove that the sequence is strictly increasing.

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

高二選擇性必修第二册(A版).pdf · 4.1 · PDF 7 / printed page 2

TOPIC 01

Concept of a sequence

Connect general terms, recurrence relations and partial sums.

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

01 / Standard#Your turn

a_n=1-\frac{12}{n+1},\quad n\ge1

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the indexed terms or partial sums before simplifying.

Hint 2

Compare consecutive terms.

Worked solution

Express the indexed terms or partial sums before simplifying.
Apply the stated relation and retain its conditions.
$a_{n+1}-a_n=12/[(n+1)(n+2)]>0$
Positive denominators and t>0 establish monotonicity for every allowed index.

The requested relation or conclusion is shown below.

a_{n+1}-a_n>0

Checks and common pitfalls: Positive denominators and t>0 establish monotonicity for every allowed index.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Defining relation: Connect general terms, recurrence relations and partial sums.
$a_n=S_n-S_{n-1}\ (n\ge2)$
Conditions: Indices are positive integers; a1=S1 needs separate treatment.