2010/059) find sqrt((48*49*50*51)+1)

we know

x(x+1)(x+2)(x+3) + 1
= (x(x+3))((x+1)(x+2)) +1
= (x^2+3x)(x^2+3x+2) + 1

put x^2+3x = t

so we get = t(t+2)+1) = t^2 + 2t + 1 = (t+1)^2
= (x^2+3x+1)^2

so sqrt(x(x+1)(x+2)(x+3)+1) = x^2+3x + 1

putting x= 48 we get
sqrt(48 * 49 * 50* 51+1) = 48^2 + 3 * 48 + 1 = 48 * 51 + 1 = 2449