Let a₁, a₂, a₃, be a strictly increasing arithmetic progression such that a₁, a₂ and a₅ form a geometric progression. If _ n=1 ^ a_n 4^n = 5 3 , then the value of a₆ is equal to

Step-by-step solution

Let the first term of the arithmetic progression be a₁ and the common difference be d . Since the AP is strictly increasing, d > 0 . The terms a₁, a₂, a₅ are a₁, a₁+d, a₁+4d . Since they form a geometric progression, (a₁+d)^2 = a₁(a₁+4d) . a₁^2 + 2a₁ d + d^2 = a₁^2 + 4a₁ d d^2 = 2a₁ d Since d > 0 , we can divide by d to get d = 2a₁ . Now, consider the infinite sum S = _ n=1 ^ a_n 4^n . This is an arithmetico-geometric progression with common ratio r = 1 4 . The sum is S = a₁ r 1-r + d r^2 (1-r)^2 . Substituting r =