To prove 1.1!+2.2!+3.3!+....................+n.n!=(n+1)!-1 using induction

To establish 1.1!+2.2!+3.3!+………………..+n.n!=(n+1)!-1 using gathering

Basic plod :

Let n=1

then 1.1!=1.1=1

and (1+1)!-1=2!-1=2-1=1

Hereafter the loving office is penny ce n=1

Gathering Plod:

Assume that office is penny ce F(k)

i.e., 1.1!+2.2!+3.3!+………+k.k!=(k+1)!-1;

Now cogitate F(k+1) = 1.1!+2.2!+3.3!+………+k.k!+(k+1)(k+1)!

= (k+1)!-1 + (k+1)(k+1)! past by overhead asumption

= (k+1)! (k+1+1) -1

= (k+1)!(k+2) -1

= (k+2)! -1 As (n+1).n = (n+1)!

Therefore hereafter establishd by F(k+1)

Hereafter by unversified gathering it is establishd that

1.1!+2.2!+3.3!+4.4!+………….+n.n! = (n+1)!-1