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Find all real geometric means between t and 4t.

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

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

Revisit first: Concept of a sequence

TOPIC 01

Geometric sequences

Connect ratios, indexed powers and finite geometric sums.

What you will be able to explain

Connect ratios, indexed powers and finite geometric sums.
Justify the method and check the conditions in a new situation.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

Find all real geometric means between t and 4t.

t=11

The geometric ratio is defined for nonzero terms; q=1 uses Sn=na1.

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.

b^2=ac

Worked solution

Use the common ratio and check its special cases.
Apply the stated relation and retain its conditions.
$b^2=11\cdot44=484$
Apply the stated relation and retain its conditions.
$b=\pm22$
Without a positivity condition, both signs are possible.

The requested relation or conclusion is shown below.

b=\pm22

Checks and common pitfalls: Without a positivity condition, both signs are possible.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

Teacher preparation and assessment

Question sequence

Connect ratios, indexed powers and finite geometric sums.
Which condition is essential in geometric sequences?
Does a negative ratio imply a decreasing sequence?

Board plan

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.

Anticipated thinking

The sum formula with denominator 1−q cannot be used directly at q=1.

Assessment checklist

1 mark: choose the correct representation and conditions.
1 mark: establish the intermediate relation.
1 mark: complete a connected calculation or proof.
1 mark: interpret and check the conclusion.