jmc

Question

Let a_1, a_2, a_3 be the first three terms of a geometric sequence. If a_1 = 1, find the smallest possible value of 4a_2 + 5a_3.

Accepted Answer

Let r be the common ratio. Then a_2 = r and a_3 = r^2, so 4a_2 + 5a_3 = 4r + 5r^2 = 5 ( r + 25 )^2 - 45.Thus, the minimum value is -45, which occurs when r = -25.