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Non-linear Recursion Problems
12-24-2010, 04:21 PM
Post: #1
Non-linear Recursion Problems
(1) Find $\{a_n\}$ such that $\displaystyle{a_1 = \frac{1}{2}, \; \sum_{k=1}^n a_k = n^2 a_n}$
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12-26-2010, 05:34 PM
Post: #2
RE: Non-linear Recursion Problems: (1)
(1) Clearly we have $(n-1)^2 a_{n-1}=(n^2-1)a_n$ and so
$\quad\quad a_n=\frac{n-1}{n+1}a_{n-1}=\frac{(n-1)(n-2)}{(n+1)n}a_{n-2}=\cdots = \frac{(n-1)(n-2)\cdots 1}{(n+1)n\cdots 3}a_1=\frac{(n-1)!}{(n+1)!}=\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$
$\quad\quad$ Check: $\sum_1^n a_i = \sum_1^n \left(\frac{1}{i}-\frac{1}{i+1} \right )=1-\frac{1}{n+1}=\frac{n}{n+1}=n^2 a_n, \quad a_1 = \frac{1}{1(1+1)}=\frac{1}{2}$
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