Use induction to prove that for all integers n 2 1 we have 1.1! +2.2! +3.3! + +nin! = (n + 1)! - 1.

Step-by-step explanation:

Let's assume that

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

For n = 1

L.H.S = 1.1!

         = 1

R.H.S = (n + 1)! - 1.

          =(1 + 1)! - 1.

          = 1

L.H.S = R.H.S

Hence the P(n) is true for n=1

Fort n = 2

L.H.S=1.1! +2.2!

        =1+4

        =5

R.H.S = (2 + 1)! - 1.

          =(2 + 1)! - 1.

          = 5

L.H.S = R.H.S

Hence the P(n) is true for n=2

Let's assume that P(n) is true for all n.

Then we have to prove that P(n) is true for (n+1) too.

So,

L.H.S = 1.1! +2.2! +3.3! + ... +n.n!+(n+1).(n+1)!

         = (n + 1)! - 1 +(n+1).(n+1)!

         = (n+1)![1+(n+1)]-1

         =(n+1)!(n+2)-1

         =(n+2)!-1

         =[(n+1)+1]!-1

So, P(n) is also true for (n+1).

So, P(n) is true for all integers n.

konichiwa~! my name is zalgo and i am here to answer your anime-related question today. more than likely, kirishima would kill izuku, marry mino and kiss momo. all of this would be my opinion by the way. (and i am also a weeb lol)

i hope this you! : d

stay and stay proud! " - zalgo

(and do you think you can mark me as brainliest? i'd greatly appreciate it! arigato~! )