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Let us consider the sequence $ a_1 , a_2 , a_3 ,\ldots $ where $ \frac{1}{a_{k+1}} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}$ for $k >1$ and $a_1 = 2^{2009}$. How can I find the value of $ a_{2011}$?

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I changed $a_1,a_2,a_3,\text{...}$ to $a_1,a_2,a_3,\ldots$ and $\frac{1}{a_2}+\text{...}+\frac{1}{a_k}$ to $\frac{1}{a_2}+\cdots+\frac{1}{a_k}$. Those are standard usage. –  Michael Hardy Nov 29 '13 at 5:23

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up vote 6 down vote accepted

Observe that $ \frac{1}{a_{k+1}} = \frac{1}{a_k} + \frac{1}{a_k}$. This will give you a simple relation between $a_k$ and $a_{k+1}$. Unfold it upto $a_1$. You have your answer.

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Thanks for your answer . –  Way to infinity Nov 29 '13 at 5:33

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