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# Any quick methods to solve?

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289
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Question: Compute $\frac{5}{4 \cdot 9} + \frac{7}{9 \cdot 16} + \frac{9}{16 \cdot 25} + \frac{11}{25 \cdot 36}$

I noted that  we can rewrite the equation as $\frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \frac{9}{4^2 \cdot 5^2} + \frac{11}{5^2 \cdot 6^2}$. How to proceed?

Jul 31, 2021

#1
+26319
+3

Question: Compute

$$\dfrac{5}{4 \cdot 9} + \dfrac{7}{9 \cdot 16} + \dfrac{9}{16 \cdot 25} + \dfrac{11}{25 \cdot 36}$$

$$\begin{array}{|rcll|} \hline && \mathbf{\dfrac{5}{4 \cdot 9} + \dfrac{7}{9 \cdot 16} + \dfrac{9}{16 \cdot 25} + \dfrac{11}{25 \cdot 36}} \\\\ &=& \dfrac{5}{9 \cdot 4} + \dfrac{7}{16 \cdot 9} + \dfrac{9}{25 \cdot 16} + \dfrac{11}{36 \cdot 25} \\\\ &=& \dfrac{9-4}{9 \cdot 4} + \dfrac{16-9}{16 \cdot 9} + \dfrac{25-16}{25 \cdot 16} + \dfrac{36-25}{36 \cdot 25} \\\\ &=&\small{ \dfrac{9}{9 \cdot 4} -\dfrac{4}{9 \cdot 4} + \dfrac{16}{16 \cdot 9} -\dfrac{9}{16 \cdot 9} + \dfrac{25}{25 \cdot 16} -\dfrac{16}{25 \cdot 16} + \dfrac{36}{36 \cdot 25} -\dfrac{25}{36 \cdot 25}} \\\\ &=& \dfrac{1}{4} -\dfrac{1}{9} + \dfrac{1}{9} -\dfrac{1}{16} + \dfrac{1}{16} -\dfrac{1}{25} + \dfrac{1}{25} -\dfrac{1}{36} \\\\ &=& \dfrac{1}{4} -\dfrac{1}{36} \\\\ &=& \dfrac{36-4}{4*36} \\\\ &=& \dfrac{32}{4*36} \\\\ &=& \dfrac{8}{36} \\\\ &=& \mathbf{\dfrac{2}{9}} \\ \hline \end{array}$$

$$\dfrac{5}{4 \cdot 9} + \dfrac{7}{9 \cdot 16} + \dfrac{9}{16 \cdot 25} + \dfrac{11}{25 \cdot 36} = \mathbf{\dfrac{2}{9}}$$

Aug 1, 2021
#2
+178
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Thank you!

xCorrosive  Aug 2, 2021