Black book 8

Algebra Level 4

x = 10 × ∑ r = 3 100 1 r 2 − 4 \large x=10 \times \sum_{r=3}^{100} \dfrac{1}{r^2-4}

Find the value of ⌊ x ⌋ \left\lfloor x \right\rfloor .


The answer is 5.

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Mehul Chaturvedi by Mehul Chaturvedi , 22, India

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2 solutions

Chew-Seong Cheong
Mar 11, 2016

x = 10 ∑ r = 3 100 1 r 2 − 4 = 10 ∑ r = 3 100 1 ( r − 2 ) ( r + 2 ) = 10 ∑ r = 3 100 1 4 ( 1 r − 2 − 1 r + 2 ) = 10 4 ( ∑ r = 1 98 1 r − ∑ r = 5 102 1 r ) = 5 2 ( 1 1 + 1 2 + 1 3 + 1 4 − 1 99 − 1 100 − 1 101 − 1 102 ) = 5.1088 ⇒ ⌊ x ⌋ = 5 \begin{aligned} x & = 10 \sum_{r=3}^{100} \frac{1}{r^2-4} \\ & = 10 \sum_{r=3}^{100} \frac{1}{(r-2)(r+2)} \\ & = 10 \sum_{r=3}^{100} \frac{1}{4} \left(\frac{1}{r-2} - \frac{1}{r+2} \right) \\ & = \frac{10}{4} \left(\sum_{r=1}^{98} \frac{1}{r} - \sum_{r=5}^{102} \frac{1}{r} \right) \\ & = \frac{5}{2} \left(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{99} - \frac{1}{100} - \frac{1}{101} - \frac{1}{102} \right) \\ & = 5.1088 \\ \Rightarrow \lfloor x \rfloor & = \boxed{5} \end{aligned}

14 Helpful 0 Interesting 1 Brilliant 0 Confused
Sridhar Sri
Mar 15, 2016

2 Helpful 0 Interesting 0 Brilliant 0 Confused

good soln. may i know which mathematics software, are you using ?

Aswin T.S. - 5 years, 3 months ago

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Daum equation editor

Sridhar Sri - 5 years, 3 months ago

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thanks.....

Aswin T.S. - 5 years, 3 months ago

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