Răspuns:
a)
[tex]\displaystyle (1+k)C_n^k=C_n^k+kC_n^k=C_n^k+k\cdot\frac{n!}{k!(n-k)!}=C_n^k+\frac{n\cdot(n-1)!}{(k-1)!(n-k)!}=\\=C_n^k+nC_{n-1}^{k-1}[/tex]
b)
Folosind egalitatea de la a) avem
[tex]1+(C_n^1+nC_{n-1}^0)+(C_n^2+nC_{n-1}^1)+\ldots+(C_n^n+nC_{n-1}^{n-1})=\\=C_n^0+C_n^1+C_n^2+\ldots+C_n^n+n(C_{n-1}^0+C_{n-1}^1+\ldots+C_{n-1}^{n-1})=\\=2^n+n2^{n-1}=2^{n-1}(n+2)[/tex]
Explicație pas cu pas:
