The sum 1× 1! + 2 × 2! + ... + 50 × 50! equals

Question

The sum 1× 1! + 2 × 2! + ... + 50 × 50! equals (A) 51 ! (B) 51 ! - 1 (C) 51 ! + 1 (D) 2 × 51!

Tardigrade Admin · Accepted Answer

1 × 1 ! +2 × 2 !+ ldots+50 × 50 ! =(2-1) 1 !+(3-1) 2 !+(4-1) 3 ! + ldots+(51-1) 50 ! =(2 !-1 !)+(3 !-2 !)+(4 !-3 !) + ldots+(51 !-50 !) =51 !-1 !=51 !-1

Q. The sum $1 \times 1! + 2 \times 2! + ... + 50 \times 50!$ equals

$51!$

$51! - 1$

$51! + 1$

$2 \times 51!$

$1 \times 1! + 2 \times 2! + \dots + 50 \times 50!$
$= (2 - 1) 1! + (3 - 1) 2! + (4 - 1) 3!$
$+ \dots + (51 - 1) 50!$
$= (2! - 1!) + (3! - 2!) + (4! - 3!)$
$+ \dots + (51! - 50!)$
$= 51! - 1! = 51! - 1$

Q. The sum 1×1!+2×2!+...+50×50! equals

51!

51!−1

51!+1

2×51!