how to prove this without mathematical induction - Split

[numeriprimi]

[FONT=&quot]Hello everyone.[/FONT]

I have to prove this equations:
a) Σ (from k=1 to n) (k²+1)*k!=n*(n+1)!
b) Σ (from k=1 to n) 2^n-k*k*(k+1)!=(n+2)!-2ⁿ⁺¹


I know how to solve it with mathematical induction. However, I can't. I have to find a different way. But I dont't see how. Please, could you tell me how to do it?

Thank you.

 

[stapel]

I have to prove these equations:

. . . . .\(\displaystyle \displaystyle \mbox{a) }\, \sum_{k=1}^n\, \left[(k^2\, +\, 1)\, \cdot\, k!\right]\, =\, n\, \cdot\, (n\, +\, 1)!\)

. . . . .\(\displaystyle \displaystyle \mbox{b) }\, \sum_{k=1}^n\, \left[2^{n-k}\, \cdot\, k\, \cdot\, (k\, +\, 1)!\right]\, =\, (n\, +\, 2)!\, -\, 2^{n+1}\)

I know how to solve it with mathematical induction. However, I can't.

Why can you not prove the relations via induction? Were there instructions or other information relating to this exercise which cause a restriction on methodology? If so, please provide the rest of the information for this exercise.

I have to find a different way. But I dont't see how.

Off the top of my head, I'm not seeing another method, either. I mean, induction was kind of invented for this exact sort of question.

How did your book and instructor approach similar exercises? Knowing how you've been shown to do these might help us figure out what's expected of you here. Thank you!