A geometric sequence has a₁=1; a₁,4a₂,16a₃ form an arithmetic sequence. Given bₙ=naₙ/4, find both general terms.

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Let q be the geometric ratio.

Hint 2

Twice the middle arithmetic term equals the outer sum.

Worked solution

Solve the repeated-root quadratic for q.
$8q=1+16q^2\iff(4q-1)^2=0\implies q=1/4$
Insert q into the two definitions.
$a_n=4^{1-n},\quad b_n=\frac n4\,4^{1-n}=\frac n{4^n}$

aₙ=4¹⁻ⁿ; bₙ=n/4ⁿ.

Checks and common pitfalls: The factors 4 and 16 apply to the sequence terms before testing arithmeticity.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Solve the repeated-root quadratic for q.
$8q=1+16q^2\iff(4q-1)^2=0\implies q=1/4$
Insert q into the two definitions.
$a_n=4^{1-n},\quad b_n=\frac n4\,4^{1-n}=\frac n{4^n}$

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

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