求助：一个递推数列问题

realnumber

\[a_1=\frac{1}{2},a_n\in R^+,a_{n+1}=\frac{a_n^2}{2}+a_n,n\in N^+,b_n=\frac{1}{a_n+2}\]
\[S_n=b_1+b_2+b_3+\cdots+b_n,T_n=b_1b_2b_3\cdots b_n\]
\[求证：2^{n+1}T_n+S_n为定值.\]

tommywong

$\displaystyle a_1=\frac{1}{2},a_{n+1}=\frac{a_n^2}{2}+a_n=\frac{1}{2}a_n(a_n+2),b_n=\frac{a_n}{2a_{n+1}}=\frac{1}{a_n}-\frac{1}{a_{n+1}}$

$\displaystyle T_n=\frac{a_1}{2^n a_{n+1}},S_n=\sum_{k=1}^n \frac{1}{a_k}-\frac{1}{a_{k+1}}=\frac{1}{a_1}-\frac{1}{a_{n+1}}$

realnumber

回复 2# tommywong


    en,thanks