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Write the sum \(\displaystyle 32+40+48 \ldots +352+360\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=40-(32)=8\). Using the first term gives the sequene \(\displaystyle a_n= 32+(n-1)(8)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 32+(n-1)(8)=360 \implies n = 42 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{42} \left(8 n + 24\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 42(32+360) }{2}=8232\).

Download \(\LaTeX\) 

\begin{question}Write the sum $32+40+48 \ldots +352+360$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=40-(32)=8$. Using the first term gives the sequene $a_n= 32+(n-1)(8)$. Setting the general term equal to the last term and solving for $n$ gives $32+(n-1)(8)=360 \implies n = 42 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{42} \left(8 n + 24\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 42(32+360) }{2}=8232$. }

\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 32+40+48 \ldots +352+360 " src="/equation_images/%20%5Cdisplaystyle%2032%2B40%2B48%20%5Cldots%20%2B352%2B360%20" alt="LaTeX:  \displaystyle 32+40+48 \ldots +352+360 " data-equation-content=" \displaystyle 32+40+48 \ldots +352+360 " />  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=40-(32)=8 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D40-%2832%29%3D8%20" alt="LaTeX:  \displaystyle a_2-a_1=40-(32)=8 " data-equation-content=" \displaystyle a_2-a_1=40-(32)=8 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 32+(n-1)(8) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2032%2B%28n-1%29%288%29%20" alt="LaTeX:  \displaystyle a_n= 32+(n-1)(8) " data-equation-content=" \displaystyle a_n= 32+(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 32+(n-1)(8)=360 \implies n = 42  " src="/equation_images/%20%5Cdisplaystyle%2032%2B%28n-1%29%288%29%3D360%20%5Cimplies%20n%20%3D%2042%20%20" alt="LaTeX:  \displaystyle 32+(n-1)(8)=360 \implies n = 42  " data-equation-content=" \displaystyle 32+(n-1)(8)=360 \implies n = 42  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{42} \left(8 n + 24\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B42%7D%20%5Cleft%288%20n%20%2B%2024%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{42} \left(8 n + 24\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{42} \left(8 n + 24\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 42(32+360) }{2}=8232 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2042%2832%2B360%29%20%7D%7B2%7D%3D8232%20" alt="LaTeX:  \displaystyle \frac{ 42(32+360) }{2}=8232 " data-equation-content=" \displaystyle \frac{ 42(32+360) }{2}=8232 " /> . </p> </p>