For a natural number n , let α n = 19 n - 12 n . Then, the value of 31 α 9 - α 10 57 α 8 is ______

Step-by-step solution

If r 1 and r 2 are roots of the quadratic a x 2 + b x + c = 0 then r 1 n = 31 r 1 n - 2 - 228 r 1 n - 1 and r 2 n = 31 r 2 n - 2 - 228 r 2 n - 1 So, r 1 n - r 2 n = 31 r 1 n - 2 - r 2 n - 2 - 228 r 1 n - 1 - r 2 n - 1 Or a n - 31 a n - 1 + 228 a n - 2 = 0 where a n = r 1 n - r 2 n Since 19   &   12 are the roots of the quadratic x 2 - 31 x + 228 = 0 we know, a n - 31 a n - 1 + 228 a n - 2 = 0 Now, for n = 10 ,    α 10 - 31 α 3 + 228 α 8 = 0 ⇒ 31 α 9 - α 10 57 &