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Question

Mathematics Question on Arithmetic Progression

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

A

51 !51\,!

B

51 !−151\,! - 1

C

51 !+151\,! + 1

D

2×51!2 \times 51!

Answer

51 !−151\,! - 1

Explanation

Solution

1×1!+2×2!+…+50×50!1 \times 1 ! +2 \times 2 !+\ldots+50 \times 50 ! =(2−1)1!+(3−1)2!+(4−1)3!=(2-1) 1 !+(3-1) 2 !+(4-1) 3 ! +…+(51−1)50!+\ldots+(51-1) 50 ! =(2!−1!)+(3!−2!)+(4!−3!)=(2 !-1 !)+(3 !-2 !)+(4 !-3 !) +…+(51!−50!)+\ldots+(51 !-50 !) =51!−1!=51!−1=51 !-1 !=51 !-1