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September 29th, 2017, 11:49 PM  #1 
Newbie Joined: Sep 2017 From: czech republic Posts: 2 Thanks: 0  how to prove this without mathematical induction
Hello everyone. a) Σ (from k=1 to n) (k^2+1)*k!=n*(n+1)! b) Σ (from k=1 to n) 2^(nk)*k*(k+1)!=(n+2)!2^(n+1) I know how to solve it with mathematical induction. However, I can't. I have to find a different way. But I don't see how. Please, could you tell me how to do it? Thank you. Last edited by skipjack; September 30th, 2017 at 12:00 AM. 
September 30th, 2017, 12:39 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,247 Thanks: 1439 
(a) If f(k) ≡ k(k + 1)!, (k² + 1)k! = f(k)  f(k  1) and f(0) = 0, so the sum is f(1)  f(0) + f(2)  f(1) + ... + f(n)  f(n  1), which is f(n). Although the above isn't set out as a use of mathematical induction, it's equivalent to the use of mathematical induction. The part (b) of your question can be tackled similarly. 
September 30th, 2017, 12:28 PM  #3 
Newbie Joined: Sep 2017 From: czech republic Posts: 2 Thanks: 0 
Thank you, now I understand. However, I don't know how to solve second equation, when the n is on both sides. It is different. How can I do it?


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