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Write the sum \(\displaystyle 44+36+28 \ldots -212-220\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=36-(44)=-8\). Using the first term gives the sequene \(\displaystyle a_n= 44+(n-1)(-8)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 44+(n-1)(-8)=-220 \implies n = 34 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{34} \left(52 - 8 n\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 34(44-220) }{2}=-2992\).

Download \(\LaTeX\) 

\begin{question}Write the sum $44+36+28 \ldots -212-220$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=36-(44)=-8$. Using the first term gives the sequene $a_n= 44+(n-1)(-8)$. Setting the general term equal to the last term and solving for $n$ gives $44+(n-1)(-8)=-220 \implies n = 34 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{34} \left(52 - 8 n\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 34(44-220) }{2}=-2992$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Write the sum  <img class="equation_image" title=" \displaystyle 44+36+28 \ldots -212-220 " src="/equation_images/%20%5Cdisplaystyle%2044%2B36%2B28%20%5Cldots%20-212-220%20" alt="LaTeX:  \displaystyle 44+36+28 \ldots -212-220 " data-equation-content=" \displaystyle 44+36+28 \ldots -212-220 " />  in sigma notation and then find the sum.</p> </p>
HTML for Canvas
<p> <p>The common difference is given by  <img class="equation_image" title=" \displaystyle a_2-a_1=36-(44)=-8 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D36-%2844%29%3D-8%20" alt="LaTeX:  \displaystyle a_2-a_1=36-(44)=-8 " data-equation-content=" \displaystyle a_2-a_1=36-(44)=-8 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 44+(n-1)(-8) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2044%2B%28n-1%29%28-8%29%20" alt="LaTeX:  \displaystyle a_n= 44+(n-1)(-8) " data-equation-content=" \displaystyle a_n= 44+(n-1)(-8) " /> . Setting the general term equal to the last term and solving for  <img class="equation_image" title=" \displaystyle n " src="/equation_images/%20%5Cdisplaystyle%20n%20" alt="LaTeX:  \displaystyle n " data-equation-content=" \displaystyle n " />  gives  <img class="equation_image" title=" \displaystyle 44+(n-1)(-8)=-220 \implies n = 34  " src="/equation_images/%20%5Cdisplaystyle%2044%2B%28n-1%29%28-8%29%3D-220%20%5Cimplies%20n%20%3D%2034%20%20" alt="LaTeX:  \displaystyle 44+(n-1)(-8)=-220 \implies n = 34  " data-equation-content=" \displaystyle 44+(n-1)(-8)=-220 \implies n = 34  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{34} \left(52 - 8 n\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B34%7D%20%5Cleft%2852%20-%208%20n%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{34} \left(52 - 8 n\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{34} \left(52 - 8 n\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 34(44-220) }{2}=-2992 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2034%2844-220%29%20%7D%7B2%7D%3D-2992%20" alt="LaTeX:  \displaystyle \frac{ 34(44-220) }{2}=-2992 " data-equation-content=" \displaystyle \frac{ 34(44-220) }{2}=-2992 " /> . </p> </p>