Respuesta :
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.
