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Determine the value of \(\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}.\)

 May 2, 2019
 #1
avatar
+1

Sum up the top and bottom separately and divide:
T = sumfor(n, 1, 2017,(2017 - n) / n) = 14495.83626
B = sumfor(n, 2, 2017,(1 / n) = 7.186830075
T / B = 14495.83626 / 7.186830075 =2017

 May 2, 2019
 #2
avatar+23903 
+6

Determine the value of

\(\large \dfrac{\dfrac{2016}{1} + \dfrac{2015}{2} + \dfrac{2014}{3} + \dots + \dfrac{1}{2016}}{\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2017}}.\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{\dfrac{2016}{1} + \dfrac{2015}{2} + \dfrac{2014}{3} + \dots + \dfrac{1}{2016}}{\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2017}} } \\\\ &=& \dfrac{\dfrac{2017-1}{1} + \dfrac{2017-2}{2} + \dfrac{2017-3}{3} + \dots + \dfrac{2017-2016}{2016}} {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2016}+ \dfrac{1}{2017}} \\\\ &=& \dfrac{\dfrac{2017}{1} + \dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016}-2016\cdot 1} {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2016}+ \dfrac{1}{2017}} \\\\ &=& \dfrac{\dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016}+2017-2016 } {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2016}+ \dfrac{1}{2017}} \\\\ &=& \dfrac{\dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016}+1 } {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2016}+ \dfrac{1}{2017}} \\\\ &=& \dfrac{\dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016} +1 } {\left( \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dots + \dfrac{1}{2016}+\dfrac{1}{2017} \right)\times \dfrac{2017}{2017} } \\\\ &=& \dfrac{\dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016} +1 } {\left( \dfrac{2017}{2} + \dfrac{2017}{3} + \dfrac{2017}{4} + \dots + \dfrac{2017}{2016}+\dfrac{2017}{2017} \right)\times \dfrac{1}{2017} } \\\\ &=& \dfrac{\dfrac{2017}{2} + \dfrac{2017}{3} + \dots + \dfrac{2017}{2016} +1 } {\left( \dfrac{2017}{2} + \dfrac{2017}{3} + \dfrac{2017}{4} + \dots + \dfrac{2017}{2016}+1 \right)\times \dfrac{1}{2017} } \\\\ &=& \dfrac{\left(\dfrac{2017}{2} + \dfrac{2017}{3}+ \dfrac{2017}{4} + \dots + \dfrac{2017}{2016} +1\right) } {\left( \dfrac{2017}{2} + \dfrac{2017}{3} + \dfrac{2017}{4} + \dots + \dfrac{2017}{2016}+1 \right) } \times 2017 \\\\ &=& \mathbf{2017} \\ \hline \end{array}\)

 

laugh

 May 3, 2019
 #3
avatar+106539 
+2

Nice, heureka   !!!!

 

 

 

cool cool cool

CPhill  May 3, 2019
 #4
avatar+23903 
+2

Thank you, CPhill !

 

laugh

heureka  May 6, 2019

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