LEARN · EXPLAIN · REVISE

Find the sum of the first t terms.

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 / Transfer#Your turn

a_n=\frac1{n(n+1)},\quad t=13

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the indexed terms or partial sums before simplifying.

Hint 2

Use this intermediate relation.

1/[n(n+1)]=1/n-1/(n+1)

Worked solution

Express the indexed terms or partial sums before simplifying.
Apply the stated relation and retain its conditions.
$S_t=(1-1/2)+(1/2-1/3)+\cdots+(1/13-1/14)$
Apply the stated relation and retain its conditions.
$S_t=1-1/14=13/14$
Retain the two boundary terms after telescoping.

The requested value is 0.928571428571.

Checks and common pitfalls: Retain the two boundary terms after telescoping.

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.