Let S be the sum of the first 9 term of the series : x + k a + x 2 + k + 2 a + x 3 + k + 4 a + x 4 + k + 6 a + … where a ≠ 0 and x ≠ 1 . If S = x 10 - x + 45 a x - 1 x - 1 , then k is equal to

Step-by-step solution

S = x + k a + x 2 + k + 2 a + x 3 + k + 4 a + x 4 + k + 6 a + … S = x + x 2 + x 3 + x 4 + . . . + k a + k + 2 a + k + 4 a + k + 6 a + … S = x x 9 - 1 x - 1 + a · 9 2 2 k + 8 2 x x 9 - 1 x - 1 + a · 9 2 2 k + 8 2 = x 10 - x + 9 · a k + 8 x - 1 x - 1 So 45 = 9 k + 8 ⇒ k = - 3