Find the general terms of both sequences.

01 / Standard#Your turn

S_n=n^2+2n,\quad b_1=2,\quad b_3=2a_4,\quad b_n\text{ geometric with }r>0

Official paper · jm01-2025 · II.1(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use aₙ=Sₙ−Sₙ₋₁ and separately check n=1.

Hint 2

Use b₃=b₁r² and the positive-ratio condition.

Worked solution

Subtract consecutive partial sums; the same formula also gives a₁=3.
$a_n=(n^2+2n)-[(n-1)^2+2(n-1)]=2n+1$
Use a₄=9 to determine the ratio.
$2r^2=18,\ r>0\implies r=3,\quad b_n=2\cdot3^{n-1}$

aₙ=2n+1; bₙ=2·3ⁿ⁻¹.

Checks and common pitfalls: The negative square root is excluded by the given positive ratio.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Subtract consecutive partial sums; the same formula also gives a₁=3.
$a_n=(n^2+2n)-[(n-1)^2+2(n-1)]=2n+1$
Use a₄=9 to determine the ratio.
$2r^2=18,\ r>0\implies r=3,\quad b_n=2\cdot3^{n-1}$

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

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