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Write the sum \(\displaystyle -10-3+4 \ldots +326+333\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=-3-(-10)=7\). Using the first term gives the sequene \(\displaystyle a_n= -10+(n-1)(7)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle -10+(n-1)(7)=333 \implies n = 50 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{50} \left(7 n - 17\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 50(-10+333) }{2}=8075\).

Download \(\LaTeX\) 

\begin{question}Write the sum $-10-3+4 \ldots +326+333$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=-3-(-10)=7$. Using the first term gives the sequene $a_n= -10+(n-1)(7)$. Setting the general term equal to the last term and solving for $n$ gives $-10+(n-1)(7)=333 \implies n = 50 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{50} \left(7 n - 17\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 50(-10+333) }{2}=8075$. }

\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 -10-3+4 \ldots +326+333 " src="/equation_images/%20%5Cdisplaystyle%20-10-3%2B4%20%5Cldots%20%2B326%2B333%20" alt="LaTeX:  \displaystyle -10-3+4 \ldots +326+333 " data-equation-content=" \displaystyle -10-3+4 \ldots +326+333 " />  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=-3-(-10)=7 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D-3-%28-10%29%3D7%20" alt="LaTeX:  \displaystyle a_2-a_1=-3-(-10)=7 " data-equation-content=" \displaystyle a_2-a_1=-3-(-10)=7 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= -10+(n-1)(7) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%20-10%2B%28n-1%29%287%29%20" alt="LaTeX:  \displaystyle a_n= -10+(n-1)(7) " data-equation-content=" \displaystyle a_n= -10+(n-1)(7) " /> . 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 -10+(n-1)(7)=333 \implies n = 50  " src="/equation_images/%20%5Cdisplaystyle%20-10%2B%28n-1%29%287%29%3D333%20%5Cimplies%20n%20%3D%2050%20%20" alt="LaTeX:  \displaystyle -10+(n-1)(7)=333 \implies n = 50  " data-equation-content=" \displaystyle -10+(n-1)(7)=333 \implies n = 50  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{50} \left(7 n - 17\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B50%7D%20%5Cleft%287%20n%20-%2017%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{50} \left(7 n - 17\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{50} \left(7 n - 17\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 50(-10+333) }{2}=8075 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2050%28-10%2B333%29%20%7D%7B2%7D%3D8075%20" alt="LaTeX:  \displaystyle \frac{ 50(-10+333) }{2}=8075 " data-equation-content=" \displaystyle \frac{ 50(-10+333) }{2}=8075 " /> . </p> </p>