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Define a sequence $$a_n=\sqrt{1+\left(1-\frac{1}{n}\right)^2}+\sqrt{1+\left(1+\frac{1}{n}\right)^2}$$ for $n \geq 1$. Find $$\sum_{i=1}^{20}\frac{1}{a_i}$$ Some insight on the approach is highly appreciated. What is the general way of solving such problems? Thanks.

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  • $\begingroup$ what do you mean by "general"? In this case here I'd probably try to figure out if one could see some sort of telescope sum or a geometric sum. May be a numerical solution is faster $\endgroup$ – user190080 Jun 24 '15 at 14:06
  • $\begingroup$ Without trying to solve it, my guess would be that if you calculate 1/ai you will end up with something trivial or with a telescope series. And reading the answers, that guess would have been right. $\endgroup$ – gnasher729 Jun 24 '15 at 16:40
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First you calculate $\frac{1}{a_i}$ and write it a bit differently:

$\frac{1}{a_n} = \frac{\sqrt{1+(1-\frac{1}{n})^2}-\sqrt{1+(1+\frac{1}{n})^2}}{(1+(1-\frac{1}{n})^2)-(1+(1+\frac{1}{n})^2)} =\frac{\sqrt{1+(1-\frac{1}{n})^2}-\sqrt{1+(1+\frac{1}{n})^2}}{-\frac{4}{n}} =\frac{1}{4} \left ( \sqrt{n^2+(n+1)^2}-\sqrt{n^2+(n-1)^2} \right )$.

Now, if you look at $\sum_{n=1}^{20} \frac{1}{4} \left ( \sqrt{n^2+(n+1)^2}-\sqrt{n^2+(n-1)^2} \right )$, you can see that it is a telescopic sum and thus can be simplified (because most of the terms cancel each other out):

$\sum_{n=1}^{20} \frac{1}{4} \left ( \sqrt{n^2+(n+1)^2}-\sqrt{n^2+(n-1)^2} \right )=\frac{1}{4} (\sqrt{20^2+21^2}-\sqrt{1^2-0^2})= 7$

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Telescoping is a useful tool.

$$\frac1{a_n} = \frac1{\sqrt{1+(1-1/n)^2}+\sqrt{1+(1+1/n)^2}} = \frac14(\sqrt{2n^2+2n+1}-\sqrt{2n^2-2n+1})$$

with $f(n) = \frac14\sqrt{2n^2+2n+1}$, this gives $\frac1{a_n} = f(n)-f(n-1)$. Can you do the rest?

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$$\frac{1}{\sqrt{x}+\sqrt{y}}=\frac{\sqrt{x}-\sqrt{y}}{x-y},$$ so by choosing $x=1+\left(1+\frac{1}{n}\right)^2$, $y=1+\left(1-\frac{1}{n}\right)^2$ we have: $$ \frac{1}{a_n} = \frac{n}{4}\left(\sqrt{1+\left(1+\frac{1}{n}\right)^2}-\sqrt{1+\left(1-\frac{1}{n}\right)^2}\right) $$ or: $$ \frac{1}{a_n} = \frac{1}{4}\left(\sqrt{n^2+(n+1)^2}-\sqrt{n^2+(n-1)^2}\right) $$ and our sum turns out to be a telescopic one: $$ \sum_{n=1}^{20}\frac{1}{a_n}=\frac{1}{4}\left(\sqrt{20^2+21^2}-1\right)=\bbox[5px,border:2px solid #C0A000]{\color{red}{7}}$$

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