LEARN · EXPLAIN · REVISE

A geometric sequence has S2=t and S4=5t. Find q².

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

高二選擇性必修第二册(A版).pdf · 4.3 · PDF 32 / printed page 27

TOPIC 01

Geometric sequences

Connect ratios, indexed powers and finite geometric sums.

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

01 / Transfer#Worked example

t=4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the common ratio and check its special cases.

Hint 2

Use this intermediate relation.

S_4=S_2(1+q^2)

Worked solution

Use the common ratio and check its special cases.
Apply the stated relation and retain its conditions.
$5=1+q^2$
Apply the stated relation and retain its conditions.
$q^2=4$
Divide by the known nonzero S2; the equation determines q² rather than the sign of q.

The requested value is 4.

Checks and common pitfalls: Divide by the known nonzero S2; the equation determines q² rather than the sign of q.

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

Teacher preparation and assessment

Defining relation: Connect ratios, indexed powers and finite geometric sums.
$a_n=a_1q^{n-1};\quad S_n=a_1(1-q^n)/(1-q)$
Conditions: The geometric ratio is defined for nonzero terms; q=1 uses Sn=na1.