For k ∈ ℕ , if the sum of the series 1 + 4 k + 8 k 2 + 13 k 3 + 19 k 4 + . . . . . . is 10 , then the value of k is

Step-by-step solution

Given, 10 = 1 + 4 k + 8 k 2 + 13 k 3 + 19 k 4 . . . . . . . . ∞   . . . . . 1 10 k = 1 k + 4 k 2 + 8 k 3 + 13 k 4 + . . . . . . . . ∞   . . . . 2 On subtracting above two equations we get, 10 - 10 k = 1 + 3 k + 4 k 2 + 5 k 3 + 6 k 4 . . . . . . . . ∞ ⇒ 10 1 - 1 k = 1 + 3 k + 4 k 2 + 5 k 3 + 6 k 4 . . . . . . . . ∞   . . . . . 3 ⇒ 10 1 - 1 k 1 k =       1 k + 3 k 2 + 4 k 3 + 5 k 4 + 6 k 5 . . . . . . . . ∞   . . . . . 4 Again subtracting bo