LEARN · EXPLAIN · REVISE

Find the constant term in (x+1/x)^2m.

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

高三選擇性必修第三册(A版).pdf · 6.3 · PDF 34 / printed page 29

TOPIC 01

Binomial theorem

Use the general term to locate coefficients, constants and coefficient sums.

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

01 / Standard#Worked example

m=7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Decide whether choices are alternative cases or successive stages, then check overlap.

Hint 2

Use this intermediate relation.

x^{2m-r}x^{-r}=x^{2m-2r}

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$14-2r=0\Rightarrow r=7$
Apply the stated relation and retain its conditions.
$T=C_{14}^{7}=3432$
A zero exponent locates the constant term; then evaluate its coefficient.

The requested value is 3432.

Checks and common pitfalls: A zero exponent locates the constant term; then evaluate its coefficient.

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

Teacher preparation and assessment

Defining relation: Use the general term to locate coefficients, constants and coefficient sums.
$(a+b)^n=\sum_{r=0}^nC_n^ra^{n-r}b^r$
Conditions: The exponent n is a nonnegative integer; term number is r+1, not r.