Find Sₙ=Σaⱼ and Tₙ=Σbⱼ for the sequences in part (a).

01 / Standard#Your turn

a_n=4^{1-n},\quad b_n=n4^{-n}

Official paper · jm01-2022 · II.2(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use a geometric sum for Sₙ.

Hint 2

Subtract Tₙ from 4Tₙ to remove the varying coefficients.

Worked solution

Compute the geometric sum.
$S_n=\frac{1-4^{-n}}{1-1/4}=\frac43(1-4^{-n})$
Shift and subtract the arithmetic-geometric sum.
$3T_n=1+\frac14+\cdots+\frac1{4^{n-1}}-\frac n{4^n}=\frac43(1-4^{-n})-n4^{-n}$
Divide by three.
$T_n=\frac49-\frac{3n+4}{9\cdot4^n}$

Sₙ=4(1−4⁻ⁿ)/3; Tₙ=4/9−(3n+4)/(9·4ⁿ).

Checks and common pitfalls: The shifted subtraction leaves a final −n/4ⁿ term.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Compute the geometric sum.
$S_n=\frac{1-4^{-n}}{1-1/4}=\frac43(1-4^{-n})$
Shift and subtract the arithmetic-geometric sum.
$3T_n=1+\frac14+\cdots+\frac1{4^{n-1}}-\frac n{4^n}=\frac43(1-4^{-n})-n4^{-n}$
Divide by three.
$T_n=\frac49-\frac{3n+4}{9\cdot4^n}$

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

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