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Write the sum \(\displaystyle 6+7+8 \ldots +36+37\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=7-(6)=1\). Using the first term gives the sequene \(\displaystyle a_n= 6+(n-1)(1)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 6+(n-1)(1)=37 \implies n = 32 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{32} \left(n + 5\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 32(6+37) }{2}=688\).

Download \(\LaTeX\) 

\begin{question}Write the sum $6+7+8 \ldots +36+37$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=7-(6)=1$. Using the first term gives the sequene $a_n= 6+(n-1)(1)$. Setting the general term equal to the last term and solving for $n$ gives $6+(n-1)(1)=37 \implies n = 32 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{32} \left(n + 5\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 32(6+37) }{2}=688$. }

\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 6+7+8 \ldots +36+37 " src="/equation_images/%20%5Cdisplaystyle%206%2B7%2B8%20%5Cldots%20%2B36%2B37%20" alt="LaTeX:  \displaystyle 6+7+8 \ldots +36+37 " data-equation-content=" \displaystyle 6+7+8 \ldots +36+37 " />  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=7-(6)=1 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D7-%286%29%3D1%20" alt="LaTeX:  \displaystyle a_2-a_1=7-(6)=1 " data-equation-content=" \displaystyle a_2-a_1=7-(6)=1 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 6+(n-1)(1) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%206%2B%28n-1%29%281%29%20" alt="LaTeX:  \displaystyle a_n= 6+(n-1)(1) " data-equation-content=" \displaystyle a_n= 6+(n-1)(1) " /> . 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 6+(n-1)(1)=37 \implies n = 32  " src="/equation_images/%20%5Cdisplaystyle%206%2B%28n-1%29%281%29%3D37%20%5Cimplies%20n%20%3D%2032%20%20" alt="LaTeX:  \displaystyle 6+(n-1)(1)=37 \implies n = 32  " data-equation-content=" \displaystyle 6+(n-1)(1)=37 \implies n = 32  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{32} \left(n + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B32%7D%20%5Cleft%28n%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{32} \left(n + 5\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{32} \left(n + 5\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 32(6+37) }{2}=688 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2032%286%2B37%29%20%7D%7B2%7D%3D688%20" alt="LaTeX:  \displaystyle \frac{ 32(6+37) }{2}=688 " data-equation-content=" \displaystyle \frac{ 32(6+37) }{2}=688 " /> . </p> </p>