Let a1, a2, and a3,... be a non constant arithmetic sequence. If a1, a4, and a8 form a geometric sequence, what is the common ratio of the geometric sequence ?

Answer 1

a_n = a₁ + (n-1)d

a₄ = a₁r = a₁ + 3d

a₈ = a₁r² = a₁ +7d

Multiply the a₄ equation by 7 and the a₈ equation by 3

7a₁r = 7a₁ + 21d

3a₁r² =3a₁ + 21d

Subtract

a₁r(7-3r) = 4a₁

and

3r² - 7r - 4 = 0

(3r-4)(r-1)=0

r=1 is useless as the ratio except in a trivial series where all the terms are the same;

therefore, r=4/3.