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Let $a_{1}>0,a_{2}>0$ and $a_{n}=\dfrac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}}, n>2$, then $\{ a_{n}\}$ converges to $\dfrac{3a_{1}a_{2}}{a_{1}+a_{2}}$.

My attempt: \begin{align} a_{n} &= \frac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}} \\ &= \frac{2}{\dfrac{1}{a_{n-1}}+\dfrac{1}{a_{n-2}}} \\ & \le \frac{1}{\sqrt{a_{n-1}a_{n-2}}} \end{align}

I used AM- GM inequality here. I am not able to proceed further. How to solve the question? Please help me.

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Hint:  let $b_n=1/a_n$ then $b_{n+1}=(b_n+b_{n-1})/2 \iff (b_{n+1}-b_n)=-(b_n-b_{n-1})/2$. Then:

$$ b_{n+1}-b_n=\frac{-1}{2}(b_n-b_{n-1})=\left(\frac{-1}{2}\right)^2(b_{n-1}-b_{n-2}) = \cdots = \left(\frac{-1}{2}\right)^{n-1}(b_{2}-b_{1}) $$

Next, telescope:

$$ b_{n+1} = b_n + \left(\frac{-1}{2}\right)^{n-1}(b_{2}-b_{1}) = b_{n-1} + \left(\left(\frac{-1}{2}\right)^{n-1}+\left(\frac{-1}{2}\right)^{n-2}\right)(b_2-b_1) = \;\cdots $$

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  • $\begingroup$ Can you please expand the answer? I am not able to think. $\endgroup$ – Unknown x Aug 10 '17 at 4:39
  • $\begingroup$ I got the answer. Your hint is really helpful $\endgroup$ – Unknown x Aug 10 '17 at 5:13
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    $\begingroup$ @M.Narayanan Thank you, and glad you worked it out. (I have always believed that a hint that you turn into a proof is more helpful and better remembered than a complete solution upfront.) $\endgroup$ – dxiv Aug 10 '17 at 5:19
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You are close.

From $a_{n} =\dfrac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}} $, $\dfrac1{a_{n}} =\dfrac{a_{n-1}+a_{n-2}}{2a_{n-1}a_{n-2}} =\dfrac12(\frac1{a_{n-2}}+\frac1{a_{n-1}}) $.

Letting $\dfrac1{a_n} =b_n $, this becomes $b_n =\frac12(b_{n-1}+b_{n-2}) $.

You should be able to solve this. (I think I did in one of my answers.)

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Following in the footsteps of @martycohen elsewhere on this page, a generalized version of the sequence in the OP can be constructed. Thus, consider

$$g_n=\frac{ag_{n-1}g_{n-2}}{bg_{n-1}+cg_{n-2}}=\frac{a}{\frac{b}{g_{n-2}}+\frac{c}{g_{n-1}}}$$

The let $f_n=1/g_n$ to find that

$$f_n=\left(\frac{c}{a} \right)f_{n-1}+\left(\frac{b}{a} \right)f_{n-2}$$

which I call a generalized Fibonacci sequence and have demonstrated solutions for arbitrary constants and initial conditions in various pages in MSE. See here, for example.

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