Question

If (x+1)(x+2)(x+3)(x+k)+1 is a perfect square, then find the value of 'k' ?
Options are as follows:
(A) 4
(B) 5
(C) 6
(D) 7

Answer 1

★ EXPONENTIAL RESOLUTION ★

$(x + 1)(x + 2)(x + 3)(x + k) + 1 = n {}^{2}$
$now \: n {}^{2} - 1 = f(x) - 1$
( n + 1 ) ( n - 1 ) = ( x + 1 ) ( x + 2) ( x+ 3 ) ( x + k )

Accordingly factors should must be resolved in consecutive sets ,
Hence , k = 4 yields the consecutive set of factors of 24 simultaneously , which further results in n² = 25