Question

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

(A)  9

(B)  10

(C)  11

(D)  12

Answer 1

Correct answer (C)

The given quadratic equation is x(x + 1)+ (x + 1)(x + 2) + ... + [x + (n - 1)](x + n) = 10n

 After simplifying, we get

Using n = 11 (where n∈I), we get

Therefore, x = -5, -6 (i.e., two consecutive integral solutions). Thus, n = 11.