LEARN · EXPLAIN · REVISE

Arithmetic sequences

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

高二選擇性必修第二册(A版).pdf · 4.2 · PDF 17 / printed page 12

Revisit first: Concept of a sequence

TOPIC 01

Arithmetic sequences

Derive terms and sums and solve indexed arithmetic models.

What you will be able to explain

Derive terms and sums and solve indexed arithmetic models.
Justify the method and check the conditions in a new situation.

Defining relation

Derive terms and sums and solve indexed arithmetic models.

a_n=a_1+(n-1)d;\quad S_n=n(a_1+a_n)/2

Conditions

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

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: How does the sign of the common difference affect the largest partial sum?

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

First six arithmetic terms: 2, 3, 4, 5, 6, 7; sum=27. The vertical display scale adjusts to include all six terms.

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

Transfer: Construct a sequence whose terms cross zero and locate its maximum partial sum.

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 a6.

a_1=2,\quad d=3

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_6=a_1+5d

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$a_6=2+15=17$
The sixth term is five differences after the first.

The requested value is 17.

Checks and common pitfalls: The sixth term is five differences after the first.

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

02 / Standard#Worked example

How many terms of 2,5,8,… are at most 3t+2?

t=3

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.

2+3(n-1)\le3t+2

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$n\le4\Rightarrow N=4$
The inclusive endpoint is counted.

The requested value is 4.

Checks and common pitfalls: The inclusive endpoint is counted.

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

03 / Transfer#Worked example

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

t=4

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.
$a_t=16$
Apply the stated relation and retain its conditions.
$S_t=4(10+16)/2=52$
The discrete row count is positive and integral; no continuous rounding is needed.

The requested value is 52.

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.

04 / Foundation#Your turn

Find a6.

a_1=5,\quad d=3

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_6=a_1+5d

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$a_6=5+15=20$
The sixth term is five differences after the first.

The requested value is 20.

Checks and common pitfalls: The sixth term is five differences after the first.

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

05 / Foundation#Your turn

Find the common difference.

a_2=6,\quad a_5=18

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.

06 / Foundation#Your turn

Find S4.

a_1=7,\quad d=2

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_4=4[2a_1+3d]/2

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$S_4=2(14+6)=40$
Average the first and last term, then multiply by the number of terms.

The requested value is 40.

Checks and common pitfalls: Average the first and last term, then multiply by the number of terms.

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

07 / Foundation#Your turn

Find a5 when S9 is given.

S_9=72

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.
$a_5=72/9=8$
With nine terms, the fifth is the middle term.

The requested value is 8.

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

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

08 / Standard#Your turn

Find a5 when S9 is given.

S_9=81

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.
$a_5=81/9=9$
With nine terms, the fifth is the middle term.

The requested value is 9.

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

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

09 / Standard#Your turn

How many terms of 2,5,8,… are at most 3t+2?

t=10

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.

2+3(n-1)\le3t+2

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$n\le11\Rightarrow N=11$
The inclusive endpoint is counted.

The requested value is 11.

Checks and common pitfalls: The inclusive endpoint is counted.

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

10 / Standard#Your turn

Find the sum of all positive terms in t,t−1,t−2,… .

t=11

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.

t+(t-1)+\cdots+1

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$N=11$
Apply the stated relation and retain its conditions.
$S=11(11+1)/2=66$
Zero does not add to the sum, and negative terms would decrease it.

The requested value is 66.

Checks and common pitfalls: Zero does not add to the sum, and negative terms would decrease it.

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

11 / Standard#Your turn

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

t=12

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.
$a_t=32$
Apply the stated relation and retain its conditions.
$S_t=12(10+32)/2=252$
The discrete row count is positive and integral; no continuous rounding is needed.

The requested value is 252.

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.

12 / Transfer#Your turn

Find the sum of all positive terms in t,t−1,t−2,… .

t=13

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.

t+(t-1)+\cdots+1

Worked solution

Use the common difference and the arithmetic-sum formula.
Apply the stated relation and retain its conditions.
$N=13$
Apply the stated relation and retain its conditions.
$S=13(13+1)/2=91$
Zero does not add to the sum, and negative terms would decrease it.

The requested value is 91.

Checks and common pitfalls: Zero does not add to the sum, and negative terms would decrease it.

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

13 / 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=14

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.
$a_t=36$
Apply the stated relation and retain its conditions.
$S_t=14(10+36)/2=322$
The discrete row count is positive and integral; no continuous rounding is needed.

The requested value is 322.

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.

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Derive terms and sums and solve indexed arithmetic models.
Which condition is essential in arithmetic sequences?
How does the sign of the common difference affect the largest partial sum?

Board plan

Defining relation: Derive terms and sums and solve indexed arithmetic models.
$a_n=a_1+(n-1)d;\quad S_n=n(a_1+a_n)/2$
Conditions: A finite sum contains n terms, hence n−1 differences.

Anticipated thinking

A middle-term identity depends on the number of terms being odd.

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.