Q. The sum $1(1!) + 2(2\ !) + 3(3!) + ... + n(n!)$ equal to
A) $3(n!) + n - 3$
B) $(n + 1)! - (n - 1)!$
C) $(n + 1)! - 1!$
D) $2(n!) - 2n - 1$
Explanation
(c)
$S_{n} = 1(1!) + 2(2!) + 3(3!) + ... + n(n!)$
$= (2 - 1(1!) + (3 - 1)(2!) + (4 - 1)(3!) + ... + \left\lbrack (n + 1) - 1 \right\rbrack(n!)$
$= (2 \bullet 1! - 1!) + (3 \bullet 2! - 2!) + (4 \bullet 3! - 3!) + ...\rbrack + \lbrack(n + 1)(n!) - (n!)$
$= (n + 1)! - 1!$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.