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tmua.me Tau Paper 1 · Q2

Paper 1Sequences & seriesAlgebra

A sequence of positive real numbers satisfies

an+2=1+an+1an(n≥1).a_{n+2}=\frac{1+a_{n+1}}{a_n}\qquad(n\ge1).

For which real values of tt are there exactly four ordered pairs (a1,a2)(a_1,a_2) such that

a1a2a3a4a5=12anda2026+a2028=t?a_1a_2a_3a_4a_5=12\qquad\text{and}\qquad a_{2026}+a_{2028}=t?
A4<t<11−434<t<11-4\sqrt3
B4<t≤11−434<t\le11-4\sqrt3
C33−3<t<11−43\sqrt{33}-3<t<11-4\sqrt3
D4≤t<11−434\le t<11-4\sqrt3
E33−3<t<4\sqrt{33}-3<t<4

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