For positive integer n, find the n maximising −5cₙ²+cₙ, where cₙ=2/aₙ and aₙ=2·3ⁿ.

01 / Standard#Your turn

Official paper · jm01-2023 · II.3(c) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Complete the square in cₙ.

Hint 2

The available values are discrete powers 3⁻ⁿ.

Worked solution

The continuous maximum is at c=1/10.
$-5c^2+c=1/20-5(c-1/10)^2$
Of the decreasing values 1/3,1/9,1/27,…, 1/9 is closest to 1/10.
$|1/9-1/10|=1/90<1/10-1/27,\quad n=2,\ f(2)=4/81$

n=2; the maximum is 4/81.

Checks and common pitfalls: The continuous optimum need not be an available sequence value.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The continuous maximum is at c=1/10.
$-5c^2+c=1/20-5(c-1/10)^2$
Of the decreasing values 1/3,1/9,1/27,…, 1/9 is closest to 1/10.
$|1/9-1/10|=1/90<1/10-1/27,\quad n=2,\ f(2)=4/81$

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

2023 JM01