2008 iTest Problems/Problem 75

Problem

Let $S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.$ Compute $\lfloor S^2\rfloor$ .

Solution

With summation notation, $S = \sum_{i=1}^{2007} \sqrt{1 + \tfrac{1}{i^2} + \tfrac{1}{(i+1)^2}}$ . By using a common denominator and simplifying, we have

$\begin{align*} S &= \sum_{i=1}^{2007} \sqrt{ \frac{i^2 (i^2 + 2i + 1) + i^2 + 2i + 1 + i^2}{i^2 (i+1)^2} } \\ &= \sum_{i=1}^{2007} \sqrt{ \frac{i^4 + 2i^3 + 3i^2 + 2i + 1}{i^2 (i+1)^2} } \\ &= \sum_{i=1}^{2007} \sqrt{ \frac{(i^2 + i + 1)^2}{i^2 (i+1)^2} } \\ &= \sum_{i=1}^{2007} \frac{i^2 + i + 1}{i^2 + i} \\ &= \sum_{i=1}^{2007} (1 + \frac{1}{i(i+1)}) \\ &= \sum_{i=1}^{2007} (1 + \frac{1}{i} - \frac{1}{i+1}) \end{align*}$

Notice that part of the terms telescope, making calculation simpler. Calculation results in $S = 2007 + 1 - \tfrac{1}{2008}$ . Thus, $S^2 = (2008 - \tfrac{1}{2008})^2 = 4032064 - 2 + (\tfrac{1}{2008})^2$ . Since $0 < (\tfrac{1}{2008})^2 < 1$ , we conclude that $\lfloor S^2\rfloor = \boxed{4032062}$ .

2008 iTest (Problems)
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2008 iTest Problems/Problem 75

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