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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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TOPIC 01

Sequences: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

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

01 / Transfer#Your turn

Find all real geometric means between t and 4t.

t=34

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.
$b^2=34\cdot136=4624$
Apply the stated relation and retain its conditions.
$b=\pm68$
Without a positivity condition, both signs are possible.

The requested relation or conclusion is shown below.

b=\pm68

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

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.