Question

Use the principle of induction to prove that

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

for all n $\in \mathbb{N}$.

Answer 1

Let P(k) be the statement that: 
1．1! + 2．2! + ... + k．k! = (k+1)! -1, where k is a positive integer. 

For k = 1, as 1! = 2! - 1, P(1) is obviously true. 

Now suppose that P(n) is true. 
Then 1．1! + 2．2! + ... + n．n! = (n+1)! -1. 

Adding (n+1)．(n+1)! to both sides, we have: 
1．1! + 2．2! + ... + n．n! + (n+1)(n+1)! 
= (n+1)! - 1 + (n+1)(n+1)! 
= (n+1)! + (n+1)(n+1)! - 1 
= (n+1)!{1 + (n+1)} - 1 
= (n+2)! - 1 

It follows that P(n+1) is true. Therefore, by induction, P(k) is true for all positive integer k. 

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Answer 2

plz refer to the attachment for answer

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