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My question is:

If $A_{n+1} = \frac{1}{1+\frac{1}{A_n}}$ ($n\in\mathbb{N}$) and $A_1=1$, then find the value of: $A_1A_2 + A_2A_3 + A_3A_4 + \cdots + A_{2010} A_{2011}.$

Please I would like to get some hints to solve this question.

  • 0
    @user1396721: You can find guides to using LaTeX [here](http://meta.stackexchange.com/questions/68388/there-should-be-universal-latex-mathjax-guide-for-sites-supporting-it/70559#70559) and [here](http://meta.math.stackexchange.com/questions/480/math-markup-diagrams-etc-pointers-please/484#484)2012-06-02

2 Answers 2

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Here is a hint: Calculate the first few values of $A_n$; you will notice a clear pattern which you can prove to be true in general with induction. Then, note that $\frac{1}{n(n+1)}=\frac{(n+1)-n}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}.$

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    Good follow-through, good technique. +12012-06-02
1

Notice that $A_{n+1} = (1+A_n^{-1})^{-1} = A_n/(1+A_n)$, we get $A_{n+1}^{-1} = 1 + A_n^{-1}$, and the recurrence relation $A_{n+1} = (\alpha{}A_n+\beta)/(\gamma{}A_n+\delta)$ where $\gamma\ne0$ can be solved systematically:

  1. Solve the equation $x = (\alpha{}x+\beta)/(\gamma{}x+\delta)$.
  2. If the equation has two distinct roots, say, $x_1$ and $x_2$, the sequence $\big\langle(A_n-x_1)/(A_n-x_2)\big\rangle_{n>0}$ is a geometric progression(AP). Goto 4.
  3. Otherwise, the equation has two same roots, say, $x_0$. The sequence $\big\langle(A_n-x_0)^{-1}\big\rangle$ is an arithmetic progression(GP).
  4. Find a closed-form for the AP or GP, then get the solution of the recurrence.

Some degenerate cases are not discussed, but they're trivial.