For each sequence in part (a), decide whether Sₙ≥12n+36 holds for a positive integer n, and find the least such n.

01 / Standard#Your turn

Official paper · jm01-2024 · II.3(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the partial sum in each case.

Hint 2

Keep the constant-sequence case separate.

Worked solution

The constant case cannot satisfy the inequality.
$S_n=3n:\quad3n\ge12n+36\iff n\le-4$
For the increasing case, solve the quadratic inequality.
$S_n=3n^2,\quad n^2-4n-12=(n-6)(n+2)\ge0\implies n\ge6$

No positive n for aₙ=3; least n=6 for aₙ=6n−3.

Checks and common pitfalls: An answer of 6 alone omits the other valid sequence.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The constant case cannot satisfy the inequality.
$S_n=3n:\quad3n\ge12n+36\iff n\le-4$
For the increasing case, solve the quadratic inequality.
$S_n=3n^2,\quad n^2-4n-12=(n-6)(n+2)\ge0\implies n\ge6$

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

2024 JM01