Find a₅+a₇ for the geometric sequence with the two given constraints.

01 / Standard#Your turn

2S_3=a_4-a_1,\quad a_3+a_5=7

Official paper · jm01-2026 · I.13 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the first terms using first term a and ratio r.

Hint 2

Factor r³−1 instead of dividing prematurely by r−1.

Worked solution

The nonzero sum ensures a is nonzero.
$2a(1+r+r^2)=a(r^3-1)\implies(r-3)(r^2+r+1)=0$
For real r, the quadratic factor is positive; shifting both terms by two multiplies the sum by r².
$r=3,\quad a_5+a_7=r^2(a_3+a_5)=9\cdot7=63$

D: 63.

Checks and common pitfalls: There is no real root of r²+r+1=0.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The nonzero sum ensures a is nonzero.
$2a(1+r+r^2)=a(r^3-1)\implies(r-3)(r^2+r+1)=0$
For real r, the quadratic factor is positive; shifting both terms by two multiplies the sum by r².
$r=3,\quad a_5+a_7=r^2(a_3+a_5)=9\cdot7=63$

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

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