Prove by mathematical induction that 3⁴ⁿ⁺²+5²ⁿ⁺¹ is divisible by 14 for every positive integer n.

01 / Standard#Your turn

Official paper · jm01-2022 · II.5 · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check n=1 explicitly.

Hint 2

Relate the n=k+1 expression to 25 times the n=k expression.

Worked solution

The base case is a multiple of 14.
$3^6+5^3=729+125=854=14\cdot61$
Assume 14 divides Aₖ=3⁴ᵏ⁺²+5²ᵏ⁺¹.
Express the next term as a sum of two multiples of 14.
$A_{k+1}=81\cdot3^{4k+2}+25\cdot5^{2k+1}=25A_k+56\cdot3^{4k+2}$
The induction hypothesis and 56=4·14 complete the inductive step, hence the claim holds for all positive n.

Divisible by 14 for every positive integer n.

Checks and common pitfalls: State both the induction hypothesis and how it proves the next case.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The base case is a multiple of 14.
$3^6+5^3=729+125=854=14\cdot61$
Assume 14 divides Aₖ=3⁴ᵏ⁺²+5²ᵏ⁺¹.
Express the next term as a sum of two multiples of 14.
$A_{k+1}=81\cdot3^{4k+2}+25\cdot5^{2k+1}=25A_k+56\cdot3^{4k+2}$
The induction hypothesis and 56=4·14 complete the inductive step, hence the claim holds for all positive n.

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

2022 JM01