The sum 1(1!) + 2(2!) + 3(3!) + ....+n (n!) equals - Vidyadip

MCQ

The sum $1(1!) + 2(2!) + 3(3!) + ....+n (n!)$ equals

A
$3\,(n\,!)\, + \,n - 3$
B
$(n + 1)!\, - \,(n - 1)!$
✓
$(n + 1)\,!\, - 1$
D
$2\,(n\,!) - 2n - 1$

✓

Answer

Correct option: C.

$(n + 1)\,!\, - 1$

c
(c) ${S_n} = 1(1!) + 2(2!) + 3(3!) + ..... + n(n!)$

=$(2 - 1)(1!) + (3 - 1)(2!) + (4 - 1)(3!) + .....$$ + [(n + 1) - 1](n!)$

= $(2.1! - 1!) + (3.2! - 2!) + (4.3! - 3!) + ....$$ + [(n + 1)(n!) - (n!)]$

=$(2! - 1!) + (3! - 2!) + (4! - 3!) + .... + [(n + 1)! - (n)!]$

= $(n + 1)! - 1!$.

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