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.