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arithmetic series

 

 Apr 17, 2020
 #1
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∑ [ (4n - 1), n , 1, 50] = 5050

 Apr 17, 2020
 #2
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arithmetic series

Evaluate:  \(\sum \limits_{n=1}^{50} (4n-1)\)
 

\(\begin{array}{|rcll|} \hline && \mathbf{\sum \limits_{n=1}^{50} (4n-1)} \\\\ &=& \sum \limits_{n=1}^{50} (4n) - \sum \limits_{n=1}^{50} (1) \\\\ &=& \sum \limits_{n=1}^{50} (4n) - 50\times 1 \\\\ &=& \sum \limits_{n=1}^{50} (4n) - 50 \\\\ &=& 4\sum \limits_{n=1}^{50} (n) - 50 \\\\ &=& 4\left(1+2+3+\ldots + 49+50 \right) - 50 \\\\ &=& 4\left( \dfrac{(1+50)}{2}\times 50 \right) - 50 \\\\ &=& 2\times 51\times 50 - 50 \\\\ &=& 5100 - 50 \\\\ &=& \mathbf{5050} \\ \hline \end{array}\)

 

 

laugh

 Apr 17, 2020

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