Question

Sirul (an)n>= 1 e dat de a1 =2 , an =an-1+nn +n , n>=2. Sa se determ an = n(n+1)(n+2)

Answer 1

[tex]a_n=a_{n-1}=n^2+n=n(n+1)\Rightarrow\\ a_n-a_{n-1}=n(n+1)\\ a_{n-1}-a_{n-2}=(n-1)n\\ ...\\ a_2-a_1=2\cdot3 \\ Adunam~ultimele~n-1~relatii~obtinand~in~memebrul~stang~suma~ \\~telescopica:\\ a_n-a_1=2\cdot3+3\cdot4+...+n(n+1)\\ a_n=1\cdot2+2\cdot3+3\cdot4+...+n(n+1)\\ a_n=1^2+1+2^2+2+..+n^2+n=\\~~~~=(1^2+2^2+...+n^2)+(1+2+3+...+n)\\ ~~~~=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}=\frac{n(n+1(n+2)}{3} [/tex]
Vezi ca ai gresit enuntul ca ai an=n(n+1)(n+2)/3