n general, if a GP has a 1st term a, and a common ratio r, then the sequence of terms is : a, ar, ar², ar³, . . ., arⁿ⁻¹.
Then, a = ar + 9 . . .so a - ar = 9 : a(1 - r) = 9 : a = 9/(1 - r)
and ar + ar² = 30 . . . so a(r + r²) = 30 : a = 30/(r + r²) . . equating the two expressions for a gives :-
9/(1 - r) = 30/(r + r²) : 9r + 9r² = 30 - 30r : 9r² + 39r - 30 = 0 : 3r² + 13r - 10 = 0 . . . this factors as (3r - 2)(r + 5) = 0.
Then r = ⅔ or r = -5 . . . if we used r = -5 then successive terms would alternate from positive to negative. As all terms are positive then r = ⅔.
Thus [using the equation a = 9/(1 - r)] : a = 9/⅓ = 27. . . . which is the required answer.
Check : 2nd term : ar = 27 x ⅔ = 18 which is 9 less than the 1st term
3rd term : ar² = 27 x (4/9) = 12 . . . and 18 + 12 = 30.
