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Write the sum \(\displaystyle 22+24+26 \ldots +72+74\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=24-(22)=2\). Using the first term gives the sequene \(\displaystyle a_n= 22+(n-1)(2)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 22+(n-1)(2)=74 \implies n = 27 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{27} \left(2 n + 20\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 27(22+74) }{2}=1296\).

Download \(\LaTeX\) 

\begin{question}Write the sum $22+24+26 \ldots +72+74$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=24-(22)=2$. Using the first term gives the sequene $a_n= 22+(n-1)(2)$. Setting the general term equal to the last term and solving for $n$ gives $22+(n-1)(2)=74 \implies n = 27 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{27} \left(2 n + 20\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 27(22+74) }{2}=1296$. }

\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 22+24+26 \ldots +72+74 " src="/equation_images/%20%5Cdisplaystyle%2022%2B24%2B26%20%5Cldots%20%2B72%2B74%20" alt="LaTeX:  \displaystyle 22+24+26 \ldots +72+74 " data-equation-content=" \displaystyle 22+24+26 \ldots +72+74 " />  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=24-(22)=2 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D24-%2822%29%3D2%20" alt="LaTeX:  \displaystyle a_2-a_1=24-(22)=2 " data-equation-content=" \displaystyle a_2-a_1=24-(22)=2 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 22+(n-1)(2) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2022%2B%28n-1%29%282%29%20" alt="LaTeX:  \displaystyle a_n= 22+(n-1)(2) " data-equation-content=" \displaystyle a_n= 22+(n-1)(2) " /> . 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 22+(n-1)(2)=74 \implies n = 27  " src="/equation_images/%20%5Cdisplaystyle%2022%2B%28n-1%29%282%29%3D74%20%5Cimplies%20n%20%3D%2027%20%20" alt="LaTeX:  \displaystyle 22+(n-1)(2)=74 \implies n = 27  " data-equation-content=" \displaystyle 22+(n-1)(2)=74 \implies n = 27  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{27} \left(2 n + 20\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B27%7D%20%5Cleft%282%20n%20%2B%2020%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{27} \left(2 n + 20\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{27} \left(2 n + 20\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 27(22+74) }{2}=1296 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2027%2822%2B74%29%20%7D%7B2%7D%3D1296%20" alt="LaTeX:  \displaystyle \frac{ 27(22+74) }{2}=1296 " data-equation-content=" \displaystyle \frac{ 27(22+74) }{2}=1296 " /> . </p> </p>