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Sequences: mixed review

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

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 / Foundation#Your turn

Find a3.

a_n=20n+1

Indices are positive integers; a1=S1 needs separate treatment.

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.

a_3=3t+1

Worked solution

Express the indexed terms or partial sums before simplifying.
Apply the stated relation and retain its conditions.
$a_3=20\cdot3+1=61$
The index is substituted in every occurrence of n.

The requested value is 61.

Checks and common pitfalls: The index is substituted in every occurrence of n.

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

02 / Foundation#Your turn

Find the common difference.

a_2=21,\quad a_5=33

A finite sum contains n terms, hence n−1 differences.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the common difference and the arithmetic-sum formula.

Hint 2

Use this intermediate relation.

a_5-a_2=3d

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$3d=12\Rightarrow d=4$
Index distance, not the larger index, counts the differences.

The requested value is 4.

Checks and common pitfalls: Index distance, not the larger index, counts the differences.

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

03 / Foundation#Your turn

Find S3.

a_1=22,\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.

S_3=a_1(1+q+q^2)

Worked solution

Use the common ratio and check its special cases.
Apply the stated relation and retain its conditions.
$S_3=22(1+3+9)=286$
Direct expansion is an independent check on the finite-sum formula.

The requested value is 286.

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.

04 / Foundation#Your turn

Prove n³−n+3tn is divisible by 3 for all n≥1.

t=23

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Verify the base case, assume the claim at k, and prove it at k+1.

Hint 2

Use this intermediate relation.

f(k+1)−f(k)=3k(k+1)+3t

Worked solution

Verify the base case, assume the claim at k, and prove it at k+1.
Apply the stated relation and retain its conditions.
$f(1)=3(23)$
Apply the stated relation and retain its conditions.
$f(k+1)=f(k)+3k(k+1)+3(23)$
Both the base and the increment are multiples of three.

The requested relation or conclusion is shown below.

3 | (n^3−n+3tn)

Checks and common pitfalls: Both the base and the increment are multiples of three.

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.

05 / Foundation#Your turn

Find a3.

a_n=24n+1

Indices are positive integers; a1=S1 needs separate treatment.

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.

a_3=3t+1

Worked solution

Express the indexed terms or partial sums before simplifying.
Apply the stated relation and retain its conditions.
$a_3=24\cdot3+1=73$
The index is substituted in every occurrence of n.

The requested value is 73.

Checks and common pitfalls: The index is substituted in every occurrence of n.

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

06 / Foundation#Your turn

Find the common difference.

a_2=25,\quad a_5=37

A finite sum contains n terms, hence n−1 differences.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the common difference and the arithmetic-sum formula.

Hint 2

Use this intermediate relation.

a_5-a_2=3d

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$3d=12\Rightarrow d=4$
Index distance, not the larger index, counts the differences.

The requested value is 4.

Checks and common pitfalls: Index distance, not the larger index, counts the differences.

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

07 / Standard#Your turn

Find S5 when q=1.

a_1=26

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.

S_5=5a_1

Worked solution

Use the common ratio and check its special cases.
Apply the stated relation and retain its conditions.
$S_5=5(26)=130$
All terms are equal; the quotient formula would have zero denominator.

The requested value is 130.

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

Prove 2ⁿ≥n+1 for every integer n≥t, using t as the base.

t=27

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Verify the base case, assume the claim at k, and prove it at k+1.

Hint 2

Use this intermediate relation.

2^(k+1)=2·2^k

Worked solution

Verify the base case, assume the claim at k, and prove it at k+1.
Apply the stated relation and retain its conditions.
$n=27:2^{27}=134217728≥28$
Apply the stated relation and retain its conditions.
$2^{k+1}≥2(k+1)≥k+2$
The explicit base check and the induction step cover only the stated domain.

The requested relation or conclusion is shown below.

2^n\ge n+1\quad(n\ge27)

Checks and common pitfalls: The explicit base check and the induction step cover only the stated domain.

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.

09 / Standard#Your turn

Find the sum of the first t terms.

a_n=\frac1{n(n+1)},\quad t=28

Indices are positive integers; a1=S1 needs separate treatment.

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.
$S_t=(1-1/2)+(1/2-1/3)+\cdots+(1/28-1/29)$
Apply the stated relation and retain its conditions.
$S_t=1-1/29=28/29$
Retain the two boundary terms after telescoping.

The requested value is 0.965517241379.

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

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

10 / Standard#Your turn

Find a5 when S9 is given.

S_9=261

A finite sum contains n terms, hence n−1 differences.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the common difference and the arithmetic-sum formula.

Hint 2

Use this intermediate relation.

S_9=9a_5

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$a_5=261/9=29$
With nine terms, the fifth is the middle term.

The requested value is 29.

Checks and common pitfalls: With nine terms, the fifth is the middle term.

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

11 / Standard#Your turn

Find a3a7 given a5=t.

t=30

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.

a_3a_7=a_5^2

Worked solution

Use the common ratio and check its special cases.
Apply the stated relation and retain its conditions.
$a_3a_7=a_1^2q^8=a_5^2=900$
Equal index sums give equal products in a nonzero geometric sequence.

The requested value is 900.

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.

12 / Standard#Your turn

Prove the scaled telescoping identity by induction.

\sum_{j=1}^n31/[j(j+1)]=31n/(n+1)

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Verify the base case, assume the claim at k, and prove it at k+1.

Hint 2

Use this intermediate relation.

S(k+1)=S(k)+t/[(k+1)(k+2)]

Worked solution

Verify the base case, assume the claim at k, and prove it at k+1.
Apply the stated relation and retain its conditions.
$n=1:31/2=31/2$
Apply the stated relation and retain its conditions.
$S_{k+1}=31k/(k+1)+31/[(k+1)(k+2)]=31(k+1)/(k+2)$
The constant factor remains in the added term and target.

The requested relation or conclusion is shown below.

S_n=31n/(n+1)

Checks and common pitfalls: The constant factor remains in the added term and target.

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

Find the sum of the first t terms.

a_n=\frac1{n(n+1)},\quad t=32

Indices are positive integers; a1=S1 needs separate treatment.

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.
$S_t=(1-1/2)+(1/2-1/3)+\cdots+(1/32-1/33)$
Apply the stated relation and retain its conditions.
$S_t=1-1/33=32/33$
Retain the two boundary terms after telescoping.

The requested value is 0.969696969697.

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

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

14 / Transfer#Your turn

A theatre has t rows; the first has 10 seats and each next row has 2 more. Find the total seats.

t=33

A finite sum contains n terms, hence n−1 differences.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the common difference and the arithmetic-sum formula.

Hint 2

Use this intermediate relation.

a_t=10+2(t-1)

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$a_t=74$
Apply the stated relation and retain its conditions.
$S_t=33(10+74)/2=1386$
The discrete row count is positive and integral; no continuous rounding is needed.

The requested value is 1386.

Checks and common pitfalls: The discrete row count is positive and integral; no continuous rounding is needed.

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

15 / 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.

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.

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Choose a foundation skill to revisit ↗

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.