If, for a positive integer n , the quadratic equation, x x + 1 + x + 1 x + 2 + . .. + x + n - 1 ¯ x + n = 10 n has two consecutive integral solutions, then n is equal to:

Step-by-step solution

On simplifying we get the quadratic equations as x 2 + x 2 + . . . + x 2 ⏟ n   times + 1 + 3 + 5 + . . . + 2 n - 1 x + 1 . 2 + 2 . 3 + . . . + n - 1 n = 10 n n x 2 + n 2 x + n n 2 - 1 3 = 10 n x 2 + n x + n 2 - 31 3 = 0 Let, α ,   β are the roots of the above equation ∴   α + β = - n ,   α β = n 2 - 31 3 Now, the difference of roots   α - β   = 1 ⇒   α - β 2 = 1 ⇒ α + β 2 - 4 α β = 1 &#86