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Write the sum \(\displaystyle 28+30+32 \ldots +106+108\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=30-(28)=2\). Using the first term gives the sequene \(\displaystyle a_n= 28+(n-1)(2)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 28+(n-1)(2)=108 \implies n = 41 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{41} \left(2 n + 26\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 41(28+108) }{2}=2788\).

Download \(\LaTeX\) 

\begin{question}Write the sum $28+30+32 \ldots +106+108$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=30-(28)=2$. Using the first term gives the sequene $a_n= 28+(n-1)(2)$. Setting the general term equal to the last term and solving for $n$ gives $28+(n-1)(2)=108 \implies n = 41 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{41} \left(2 n + 26\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 41(28+108) }{2}=2788$. }

\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 28+30+32 \ldots +106+108 " src="/equation_images/%20%5Cdisplaystyle%2028%2B30%2B32%20%5Cldots%20%2B106%2B108%20" alt="LaTeX:  \displaystyle 28+30+32 \ldots +106+108 " data-equation-content=" \displaystyle 28+30+32 \ldots +106+108 " />  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=30-(28)=2 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D30-%2828%29%3D2%20" alt="LaTeX:  \displaystyle a_2-a_1=30-(28)=2 " data-equation-content=" \displaystyle a_2-a_1=30-(28)=2 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 28+(n-1)(2) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2028%2B%28n-1%29%282%29%20" alt="LaTeX:  \displaystyle a_n= 28+(n-1)(2) " data-equation-content=" \displaystyle a_n= 28+(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 28+(n-1)(2)=108 \implies n = 41  " src="/equation_images/%20%5Cdisplaystyle%2028%2B%28n-1%29%282%29%3D108%20%5Cimplies%20n%20%3D%2041%20%20" alt="LaTeX:  \displaystyle 28+(n-1)(2)=108 \implies n = 41  " data-equation-content=" \displaystyle 28+(n-1)(2)=108 \implies n = 41  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{41} \left(2 n + 26\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B41%7D%20%5Cleft%282%20n%20%2B%2026%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{41} \left(2 n + 26\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{41} \left(2 n + 26\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 41(28+108) }{2}=2788 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2041%2828%2B108%29%20%7D%7B2%7D%3D2788%20" alt="LaTeX:  \displaystyle \frac{ 41(28+108) }{2}=2788 " data-equation-content=" \displaystyle \frac{ 41(28+108) }{2}=2788 " /> . </p> </p>