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Write the sum \(\displaystyle 44+53+62 \ldots +296+305\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=53-(44)=9\). Using the first term gives the sequene \(\displaystyle a_n= 44+(n-1)(9)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 44+(n-1)(9)=305 \implies n = 30 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{30} \left(9 n + 35\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 30(44+305) }{2}=5235\).

Download \(\LaTeX\) 

\begin{question}Write the sum $44+53+62 \ldots +296+305$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=53-(44)=9$. Using the first term gives the sequene $a_n= 44+(n-1)(9)$. Setting the general term equal to the last term and solving for $n$ gives $44+(n-1)(9)=305 \implies n = 30 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{30} \left(9 n + 35\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 30(44+305) }{2}=5235$. }

\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+53+62 \ldots +296+305 " src="/equation_images/%20%5Cdisplaystyle%2044%2B53%2B62%20%5Cldots%20%2B296%2B305%20" alt="LaTeX:  \displaystyle 44+53+62 \ldots +296+305 " data-equation-content=" \displaystyle 44+53+62 \ldots +296+305 " />  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=53-(44)=9 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D53-%2844%29%3D9%20" alt="LaTeX:  \displaystyle a_2-a_1=53-(44)=9 " data-equation-content=" \displaystyle a_2-a_1=53-(44)=9 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 44+(n-1)(9) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2044%2B%28n-1%29%289%29%20" alt="LaTeX:  \displaystyle a_n= 44+(n-1)(9) " data-equation-content=" \displaystyle a_n= 44+(n-1)(9) " /> . 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)(9)=305 \implies n = 30  " src="/equation_images/%20%5Cdisplaystyle%2044%2B%28n-1%29%289%29%3D305%20%5Cimplies%20n%20%3D%2030%20%20" alt="LaTeX:  \displaystyle 44+(n-1)(9)=305 \implies n = 30  " data-equation-content=" \displaystyle 44+(n-1)(9)=305 \implies n = 30  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{30} \left(9 n + 35\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B30%7D%20%5Cleft%289%20n%20%2B%2035%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{30} \left(9 n + 35\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{30} \left(9 n + 35\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 30(44+305) }{2}=5235 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2030%2844%2B305%29%20%7D%7B2%7D%3D5235%20" alt="LaTeX:  \displaystyle \frac{ 30(44+305) }{2}=5235 " data-equation-content=" \displaystyle \frac{ 30(44+305) }{2}=5235 " /> . </p> </p>