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

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

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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.

Try it. Leave your reasoning visible.

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

01 / 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.

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.