perplexus.info :: Sequences : Factorials galore

Factorials galore (Posted on 2023-04-04)

Given

S(n)= 1*1!+2*2!+3*3!+4*4!+…n*(n)!

Evaluate S(100)

No Solution Yet Submitted by Ady TZIDON

Rating: 5.0000 (1 votes)

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Puzzle Solution

| Comment 1 of 2

S(n)= 1*1!+2*2!+3*3!+4*4!+......+n*n!

= (2-1)*1! +(3-1)*2! + (4-1)*3!+.......+(n+1-1)*n!

= 2*1! -1*1! +3*2!- 1*2! + 4*3! - 3! + ......+ (n+1)*n! -1*n!

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

= (n+1)! -1

Therefore, S(100) = 101! -1

Edited on April 4, 2023, 1:24 pm

Posted by K Sengupta on 2023-04-04 13:12:36

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