Question

Sumă de la k=1 până la n din k*k! = ?​

Answer 1

k*k! = (k+1-1)*k! = (k+1)*k! - k! = (k+1)! - k!

Scriem suma astfel (este o suma telescopica) :

2! - 1!

3! - 2!

4! - 3!

...

(n+1)! - n!

-------------

Se reduc termenii pe diagonala ⇒ S = (n+1)! - 1

Answer 2

Salut,

$\displaystyle S=\sum\linits_{k=1}^nk\cdot k!=\sum\linits_{k=1}^n (k+1-1)\cdot k!=\\\\=\sum\linits_{k=1}^n(k+1)\cdot k!-k!=\sum\linits_{k=1}^n(k+1)!-k!=\\\\=2!-1!+\\\\+3!-2!+\\\\+\ldots+\\\\+n!-(n-1)!+\\\\+(n+1)!-n!=(n+1)!-1!=(n+1)!-1,\ deci\ \boxed{\ S=(n+1)!-1.\ }$

Ai înțeles ?

Green eyes.