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Mathematical induction

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

高二選擇性必修第二册(A版).pdf · 4.4 · PDF 49 / printed page 44

Revisit first: Arithmetic sequences Geometric sequences

TOPIC 01

Mathematical induction

Build proofs with a verified base, an explicit induction hypothesis and a valid step.

What you will be able to explain

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Justify the method and check the conditions in a new situation.

Defining relation

Build proofs with a verified base, an explicit induction hypothesis and a valid step.

P(n_0)\land[P(k)\Rightarrow P(k+1)]

Conditions

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

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can a correct induction step rescue a false or missing base case?

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

For n=4, the odd-number sum equals n². The false formula n²+1 has exactly the same increment 2n+1 but fails its base case when c≠0. Finite checks alone are not an induction proof.

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

Transfer: Identify the failed link in a proposed proof and repair the domain or argument.

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

Prove the shifted-integer sum by induction.

\sum_{j=1}^n(j+2)=n(2)+n(n+1)/2

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)+k+1+t

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.
$n=1:1+2=2+1$
Apply the stated relation and retain its conditions.
$S_{k+1}=2k+k(k+1)/2+k+1+2$
Apply the stated relation and retain its conditions.
$S_{k+1}=2(k+1)+(k+1)(k+2)/2$
The next shifted term includes the fixed constant.

The requested relation or conclusion is shown below.

S_n=2n+n(n+1)/2

Checks and common pitfalls: The next shifted term includes the fixed constant.

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.

02 / Standard#Worked example

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

t=3

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.
$n=3:2^{3}=8≥4$
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\ge3)

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.

03 / Transfer#Worked example

Disprove the claim that n²+n+t is prime for every nonnegative integer n.

t=4

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

Find one valid counterexample.

Hint 2

Try n=t.

Worked solution

Find one valid counterexample.
Apply the stated relation and retain its conditions.
$n=4 ⇒ n^2+n+t=4(6)$
Both factors exceed one, so the expression is composite.

The requested relation or conclusion is shown below.

n=4:n^2+n+t=4·6

Checks and common pitfalls: Both factors exceed one, so the expression is composite.

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.

04 / Foundation#Your turn

Prove the shifted-integer sum by induction.

\sum_{j=1}^n(j+5)=n(5)+n(n+1)/2

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)+k+1+t

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.
$n=1:1+5=5+1$
Apply the stated relation and retain its conditions.
$S_{k+1}=5k+k(k+1)/2+k+1+5$
Apply the stated relation and retain its conditions.
$S_{k+1}=5(k+1)+(k+1)(k+2)/2$
The next shifted term includes the fixed constant.

The requested relation or conclusion is shown below.

S_n=5n+n(n+1)/2

Checks and common pitfalls: The next shifted term includes the fixed constant.

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

Prove the odd-number sum identity.

1+3+\cdots+(2n-1)=n^2

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+2k+1

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.
$n=1:\ 1=1$
Apply the stated relation and retain its conditions.
$S_{k+1}=k^2+2k+1=(k+1)^2$
The newly added odd term is 2k+1, not 2k−1.

The requested relation or conclusion is shown below.

S_n=n^2

Checks and common pitfalls: The newly added odd term is 2k+1, not 2k−1.

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.

06 / Foundation#Your turn

Prove the finite power-of-two identity.

1+2+\cdots+2^{n-1}=2^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+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.
$n=1:\ 1=2-1$
Apply the stated relation and retain its conditions.
$S_{k+1}=2^k-1+2^k=2^{k+1}-1$
At n terms the highest exponent is n−1.

The requested relation or conclusion is shown below.

S_n=2^n-1

Checks and common pitfalls: At n terms the highest exponent is n−1.

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.

07 / Foundation#Your turn

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

t=8

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.
$f(1)=3(8)$
Apply the stated relation and retain its conditions.
$f(k+1)=f(k)+3k(k+1)+3(8)$
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.

08 / Standard#Your turn

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

t=9

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.
$f(1)=3(9)$
Apply the stated relation and retain its conditions.
$f(k+1)=f(k)+3k(k+1)+3(9)$
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.

09 / Standard#Your turn

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

t=10

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.
$n=10:2^{10}=1024≥11$
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\ge10)

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.

10 / Standard#Your turn

Prove the scaled telescoping identity by induction.

\sum_{j=1}^n11/[j(j+1)]=11n/(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.
$n=1:11/2=11/2$
Apply the stated relation and retain its conditions.
$S_{k+1}=11k/(k+1)+11/[(k+1)(k+2)]=11(k+1)/(k+2)$
The constant factor remains in the added term and target.

The requested relation or conclusion is shown below.

S_n=11n/(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.

11 / Standard#Your turn

Disprove the claim that n²+n+t is prime for every nonnegative integer n.

t=12

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

Find one valid counterexample.

Hint 2

Try n=t.

Worked solution

Find one valid counterexample.
Apply the stated relation and retain its conditions.
$n=12 ⇒ n^2+n+t=12(14)$
Both factors exceed one, so the expression is composite.

The requested relation or conclusion is shown below.

n=12:n^2+n+t=12·14

Checks and common pitfalls: Both factors exceed one, so the expression is composite.

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.

12 / Transfer#Your turn

Prove the scaled telescoping identity by induction.

\sum_{j=1}^n13/[j(j+1)]=13n/(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.
$n=1:13/2=13/2$
Apply the stated relation and retain its conditions.
$S_{k+1}=13k/(k+1)+13/[(k+1)(k+2)]=13(k+1)/(k+2)$
The constant factor remains in the added term and target.

The requested relation or conclusion is shown below.

S_n=13n/(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

Disprove the claim that n²+n+t is prime for every nonnegative integer n.

t=14

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

Find one valid counterexample.

Hint 2

Try n=t.

Worked solution

Find one valid counterexample.
Apply the stated relation and retain its conditions.
$n=14 ⇒ n^2+n+t=14(16)$
Both factors exceed one, so the expression is composite.

The requested relation or conclusion is shown below.

n=14:n^2+n+t=14·16

Checks and common pitfalls: Both factors exceed one, so the expression is composite.

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

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Teacher preparation and assessment

Question sequence

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Which condition is essential in mathematical induction?
Can a correct induction step rescue a false or missing base case?

Board plan

Defining relation: Build proofs with a verified base, an explicit induction hypothesis and a valid step.
$P(n_0)\land[P(k)\Rightarrow P(k+1)]$
Conditions: Induction proves a statement only for the specified integer domain starting at the base case.

Anticipated thinking

Checking many examples does not establish the induction step.

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.