An arithmetic sequence starts at 3, and a₁,a₂,a₅ form a geometric sequence. Find every possible general term.

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write a₂ and a₅ using the common difference d.

Hint 2

The middle geometric term squared equals the outer product.

Worked solution

Form and factor the equation.
$(3+d)^2=3(3+4d)\iff d(d-6)=0$
Retain both valid common differences.
$d=0\implies a_n=3;\quad d=6\implies a_n=6n-3$

aₙ=3 or aₙ=6n−3.

Checks and common pitfalls: A constant sequence is also both arithmetic and geometric.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Form and factor the equation.
$(3+d)^2=3(3+4d)\iff d(d-6)=0$
Retain both valid common differences.
$d=0\implies a_n=3;\quad d=6\implies a_n=6n-3$

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

2024 JM01