Prove the recurrence formula by induction and give f(xₙ), as requested by the printed wording.

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Prove the recurrence formula by induction and give f(xₙ), as requested by the printed wording.

x_1=\frac32,\quad x_{n+1}=\frac{3x_n}{x_n+3}

Official paper · jm01-2026 · II.5(a)(ii) · PDF 5

Official original and suggested answers ↗ · Suggested answer PDF page 8

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The first three terms suggest xₙ=3/(n+1).

Hint 2

Distinguish the nth term from its image under f.

Worked solution

The base case n=1 gives 3/2, as required.
$x_1=\frac3{1+1}$
Assume the formula for k, then apply the recurrence.
$x_{k+1}=\frac{3\cdot3/(k+1)}{3/(k+1)+3}=\frac3{k+2}$
Thus induction establishes xₙ; applying f increases the index by one.
$x_n=\frac3{n+1},\quad f(x_n)=x_{n+1}=\frac3{n+2}\quad(n\ge1)$

xₙ=3/(n+1); therefore f(xₙ)=3/(n+2).

Checks and common pitfalls: The printed question asks for f(xₙ), whereas the suggested answer proves xₙ. Both are shown to avoid an off-by-one error.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The base case n=1 gives 3/2, as required.
$x_1=\frac3{1+1}$
Assume the formula for k, then apply the recurrence.
$x_{k+1}=\frac{3\cdot3/(k+1)}{3/(k+1)+3}=\frac3{k+2}$
Thus induction establishes xₙ; applying f increases the index by one.
$x_n=\frac3{n+1},\quad f(x_n)=x_{n+1}=\frac3{n+2}\quad(n\ge1)$

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

Prove the recurrence formula by induction and give f(xₙ), as requested by the printed wording.

2026 JM01

Try it. Leave your reasoning visible.

Prove the recurrence formula by induction and give f(xₙ), as requested by the printed wording.