Factorial Sum!

Can be written as (2-1) 1! + (3-1) 2! + ...... (25-1) 24! + (26-1) 25! Further, it simplifies to 2!-1!+3!-2!+4!-3!+.....25!-24!+26!-25! which gives us 26!-1 as the correct answer.

We can prove by induction that, more generally:

$\sum_{k=1}^n k\cdot k! = (n+1)!-1$

for all positive integers $n$ .

For $n=1$ we get $1=2!-1$ , which is correct.

Suppose now that the statement holds for certain $n\in\mathbb{N}$ . Then we have:

$\sum_{k=1}^{n+1} k\cdot k! = (n+1)!-1+(n+1)(n+1)! = (n+2)(n+1)!-1 = (n+2)!-1.$

Hence the statement holds for all natural $n$ . Plugging in $n=25$ , we obtain:

$1\cdot 1!+2\cdot 2!+\ldots+25\cdot 25!=\boxed{26!-1}.$

Let $S=1\cdot 1!+2\cdot 2!+3\cdot 3!+...25\cdot 25!$ and let $A=1!+2!+3!+4!+...25!$ . Then,

$S+A=(1\cdot 1!+1!) +(2\cdot 2!+2!) +(3\cdot 3!+3!) +...(25\cdot 25!+25!)$ Or, $S+A= 2!+3!+4!+...26! = A-1 + 26!$

This gives,

$S+A=26!+A-1$

Or, $S = 26!-1$ .

the LHS can be written as (1-0).1!+(3-1).2!+(4-1).3!+.......+(26-1).25! on simplification we will get 1!+3!-2!+4!-3!+5!-4!...........+26!-25! = 1!-2!+3!-3!+4!-4!+5!-5!+..........+25!-25!+26! =26!-2+1 = 26!-1

k*k!=(k+1)!-k! so all of the terms cancels out except of -1! and 26!, Which gives us 26!-1!

nth term = n.n! In place of first n put n+1-1 and multiply. nth term becomes n+1! - n!.

Can be proven with a strong induction: $(n+1)!-1+n+1(n+1)! = (n+1)!(n+1+1)-1 = (n+2)!-1$

I did something totally different from all the other comments here.

what i did is that i found out what the answer of 1 1!+2 2!+3 3!+4 4!+5*5! would be.

i basically then solved 1+4+18+36+600, and got 719.

i then noticed that my answer is just 1 number away from 6!, and thus, my answer = 6!-1.

I applied this for the full question, and immediately saw that the correct answer is 26!-1.

I have been using this technique for several problems here, and it always works. reduce the problem to smaller numbers which are easier to deal with, calculate them, and then predict what the answer is going to be. For some problems you will need more calculations to predict the answer even better.

may not be the best "mathematical way", but this technique could be useful when stuck at questions in competitive maths.

let 1·1!+2·2!+3·3!+.........+25·25! = x

Add 1!+2!+3!+.......+25! on both sides

we get,

  2·1!+3·2!+4·3!+...........+26·25! = x + 1!+2!+3!+..........+25!

  2!  + 3! + 4! + .........+25! + 26! = x + 1! + 2! + 3! +.......... +25!

  26! = x + 1!

Thus, x=26! - 1! = 26! -1

or,

1·1!+2·2!+3·3!+.........+25·25! = x = 26! - 1

It's a formula. The proof is simple. Replace n by n+1-1 in n.n! to get n+1!-n! Then simply add the terms to get n+1!-1.Therefore, the answer 26!-1