Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Geometric sequences

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

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.

Defining relation

Connect ratios, indexed powers and finite geometric sums.

an=a1qn−1;Sn=a1(1−qn)/(1−q)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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does a negative ratio imply a decreasing sequence?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

First six geometric terms: 2, 2, 2, 2, 2, 2; sum=12. The vertical display scale adjusts to include all six terms.

First six geometric terms: 2, 2, 2, 2, 2, 2; sum=12. The vertical display scale adjusts to include all six terms.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare term signs and partial sums for positive and negative ratios.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find a4.

a1=2,q=2a_1=2,\quad q=2
  • 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.
a4=a1q3a_4=a_1q^3
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    a4=2⋅8=16a_4=2\cdot8=16

  3. Use n−1 in the exponent.

The requested value is 16.

Checks and common pitfalls: Use n−1 in the exponent.

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

02 / Standard#Worked example

Find a3a7 given a5=t.

t=3t=3
  • 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.
a3a7=a52a_3a_7=a_5^2
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    a3a7=a12q8=a52=9a_3a_7=a_1^2q^8=a_5^2=9

  3. Equal index sums give equal products in a nonzero geometric sequence.

The requested value is 9.

Checks and common pitfalls: Equal index sums give equal products in a nonzero geometric sequence.

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

03 / Transfer#Worked example

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

t=4t=4
  • 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.
S4=S2(1+q2)S_4=S_2(1+q^2)
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    5=1+q25=1+q^2

  3. Apply the stated relation and retain its conditions.

    q2=4q^2=4

  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.

04 / Foundation#Your turn

Find a4.

a1=5,q=2a_1=5,\quad q=2
  • 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.
a4=a1q3a_4=a_1q^3
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    a4=5⋅8=40a_4=5\cdot8=40

  3. Use n−1 in the exponent.

The requested value is 40.

Checks and common pitfalls: Use n−1 in the exponent.

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

05 / Foundation#Your turn

Find q.

a2=6,a3=−12a_2=6,\quad a_3=-12
  • 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.
q=a3/a2q=a_3/a_2
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    q=−2q=-2

  3. The sign of the ratio controls alternating signs.

The requested value is -2.

Checks and common pitfalls: The sign of the ratio controls alternating signs.

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

06 / Foundation#Your turn

Find S3.

a1=7,q=3a_1=7,\quad q=3
  • 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.
S3=a1(1+q+q2)S_3=a_1(1+q+q^2)
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    S3=7(1+3+9)=91S_3=7(1+3+9)=91

  3. Direct expansion is an independent check on the finite-sum formula.

The requested value is 91.

Checks and common pitfalls: Direct expansion is an independent check on the finite-sum formula.

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

07 / Foundation#Your turn

Find S5 when q=1.

a1=8a_1=8
  • 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.
S5=5a1S_5=5a_1
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    S5=5(8)=40S_5=5(8)=40

  3. All terms are equal; the quotient formula would have zero denominator.

The requested value is 40.

Checks and common pitfalls: All terms are equal; the quotient formula would have zero denominator.

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

08 / Standard#Your turn

Find S5 when q=1.

a1=9a_1=9
  • 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.
S5=5a1S_5=5a_1
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    S5=5(9)=45S_5=5(9)=45

  3. All terms are equal; the quotient formula would have zero denominator.

The requested value is 45.

Checks and common pitfalls: All terms are equal; the quotient formula would have zero denominator.

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

09 / Standard#Your turn

Find a3a7 given a5=t.

t=10t=10
  • 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.
a3a7=a52a_3a_7=a_5^2
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    a3a7=a12q8=a52=100a_3a_7=a_1^2q^8=a_5^2=100

  3. Equal index sums give equal products in a nonzero geometric sequence.

The requested value is 100.

Checks and common pitfalls: Equal index sums give equal products in a nonzero geometric sequence.

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

10 / Standard#Your turn

Find all real geometric means between t and 4t.

t=11t=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.
b2=acb^2=ac
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    b2=11⋅44=484b^2=11\cdot44=484

  3. Apply the stated relation and retain its conditions.

    b=±22b=\pm22

  4. Without a positivity condition, both signs are possible.

The requested relation or conclusion is shown below.

b=±22b=\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.

11 / Standard#Your turn

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

t=12t=12
  • 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.
S4=S2(1+q2)S_4=S_2(1+q^2)
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    5=1+q25=1+q^2

  3. Apply the stated relation and retain its conditions.

    q2=4q^2=4

  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.

12 / Transfer#Your turn

Find all real geometric means between t and 4t.

t=13t=13
  • 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.
b2=acb^2=ac
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    b2=13⋅52=676b^2=13\cdot52=676

  3. Apply the stated relation and retain its conditions.

    b=±26b=\pm26

  4. Without a positivity condition, both signs are possible.

The requested relation or conclusion is shown below.

b=±26b=\pm26

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.

13 / Transfer#Your turn

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

t=14t=14
  • 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.
S4=S2(1+q2)S_4=S_2(1+q^2)
Worked solution

  1. Use the common ratio and check its special cases.

  2. Apply the stated relation and retain its conditions.

    5=1+q25=1+q^2

  3. Apply the stated relation and retain its conditions.

    q2=4q^2=4

  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.

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    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.
      an=a1qn−1;Sn=a1(1−qn)/(1−q)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.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗