Întrebare Combinatorica

Buna ziua
Valoarea sumei:
\[S_n=k+\dfrac{k^2\cdot C_{n}^{1}}{2}+\dfrac{k^3\cdot C_{n}^{2}}{3}+\dots+\dfrac{k^{n+1}\cdot C_{n}^{n}}{n+1}\\ pentru\ k\in N\ fixat\ este: a)S_n=k(n+1);b)1\\ c)S_n=\dfrac{k^n-1}{n+1};d)S_n=\dfrac{k^{n+1}-1}{n+1}\\ e)S_n=\dfrac{(k+1)^{n+1}-1}{n+1}\]

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Buna ziua,

Folositi dezvoltarea: \(\left ( 1+x \right )^{n}=1+xC_{n}^{1}+x^{2}C_{n}^{2}+...+x^{n}C_{n}^{n},\: x\in \mathbb{R}\).

Ce s-ar intampla daca ati integra aceasta relatie?

Care este raspunsul corect?

Buna ziua
Pai daca integram avem:
\[\int_0^k(1+x)^ndx=x+\frac{x^2}{2}C_{n}^{1}+\dfrac{x^3}{3}C_{n}^2+.....+\dfrac{x^{n+1}}{n+1}C_n^n\\ =\dfrac{(1+x)^{n+1}}{n+1}|_0^k=\dfrac{(1+k)^{n+1}}{n+1}-\dfrac{1}{n+1}=\dfrac{(1+k)^{n+1}-1}{n+1}\]
raspuns corect (e)
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