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$\frac{1}{{102 \times 101 \times 100}} + \frac{1}{{101 \times 100 \times 99}} + ......... + \frac{1}{{3 \times 2 \times 1}} = ?$

On rewriting we have the series as,

$\frac{1}{{1 \times 2 \times 3}} + \frac{1}{{2 \times 3 \times 4}} + ................. + \frac{1}{{100 \times 101 \times 102}}$

$ = \sum\limits_{r = 1}^{100} {\frac{1}{{r(r + 1)(r + 2)}}} $

$ = \frac{1}{2}\sum\limits_{r = 1}^{100} {\left[ {\frac{1}{{r(r + 1)}} - \frac{1}{{(r + 1)(r + 2)}}} \right]} $

$ = \frac{1}{2}\left[ {\left( {\frac{1}{{1 \times 2}} - \frac{1}{{2 \times 3}}} \right) + \left( {\frac{1}{{2 \times 3}} - \frac{1}{{3 \times 4}}} \right) + .............+\left( {\frac{1}{{100 \times 101}} - \frac{1}{{101 \times 102}}} \right)} \right]$

$ = \frac{1}{2}\left( {\frac{1}{{1 \times 2}} - \frac{1}{{101 \times 102}}} \right)$

$ = \frac{1}{4}\left( {1 - \frac{1}{{101 \times 51}}} \right)$

$ = \frac{1}{4}\left( {\frac{{101 \times 51 - 1}}{{101 \times 51}}} \right)$

$ = \frac{1}{4} \times \frac{{5150}}{{101 \times 51}} = \frac{{2575}}{{2 \times 101 \times 51}} = \frac{{2575}}{{10302}}$

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${\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$

The range of the function $f(x) = {\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$ is: (A) $[ - 2,2]$ (B) $\left[ {\frac{1}{{\sqrt 5 }},\sqrt 5 } \right]$ (C) $(0,\sqrt 5 )$ (D) $[ 0,2]$ Solution We have, $f(x) = {\log _{\sqrt 5 }}\left( {3 - 2\sin \frac{{3\pi }}{4}\sin x + 2\cos \frac{\pi }{4}\cos x} \right)$ $ \Rightarrow f(x) = {\log _{\sqrt 5 }}\left[ {3 + \sqrt 2 (\cos x - \sin x)} \right]$ Now, $ - \sqrt 2  \le \cos x - \sin x \le \sqrt 2 $ $\therefore - 2 \le \sqrt 2 (\cos x - \sin x) \le 2$ $\therefore 1 \le 3 + \sqrt 2 (\cos x - \sin x) \le 5$ $\therefore{\log _{\sqrt 5 }}1 \le {\log _{\sqrt 5 }}[3 + \sqrt 2 (\cos x - \sin x)] \le {\log _{\sqrt 5 }}5$ $ \Rightarrow 0 \le f(x) \le 2$ Answer: (D)
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