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Write the sum \(\displaystyle -7-5-3 \ldots +91+93\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=-5-(-7)=2\). Using the first term gives the sequene \(\displaystyle a_n= -7+(n-1)(2)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle -7+(n-1)(2)=93 \implies n = 51 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{51} \left(2 n - 9\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 51(-7+93) }{2}=2193\).

Download \(\LaTeX\) 

\begin{question}Write the sum $-7-5-3 \ldots +91+93$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=-5-(-7)=2$. Using the first term gives the sequene $a_n= -7+(n-1)(2)$. Setting the general term equal to the last term and solving for $n$ gives $-7+(n-1)(2)=93 \implies n = 51 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{51} \left(2 n - 9\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 51(-7+93) }{2}=2193$. }

\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 -7-5-3 \ldots +91+93 " src="/equation_images/%20%5Cdisplaystyle%20-7-5-3%20%5Cldots%20%2B91%2B93%20" alt="LaTeX:  \displaystyle -7-5-3 \ldots +91+93 " data-equation-content=" \displaystyle -7-5-3 \ldots +91+93 " />  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=-5-(-7)=2 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D-5-%28-7%29%3D2%20" alt="LaTeX:  \displaystyle a_2-a_1=-5-(-7)=2 " data-equation-content=" \displaystyle a_2-a_1=-5-(-7)=2 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= -7+(n-1)(2) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%20-7%2B%28n-1%29%282%29%20" alt="LaTeX:  \displaystyle a_n= -7+(n-1)(2) " data-equation-content=" \displaystyle a_n= -7+(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 -7+(n-1)(2)=93 \implies n = 51  " src="/equation_images/%20%5Cdisplaystyle%20-7%2B%28n-1%29%282%29%3D93%20%5Cimplies%20n%20%3D%2051%20%20" alt="LaTeX:  \displaystyle -7+(n-1)(2)=93 \implies n = 51  " data-equation-content=" \displaystyle -7+(n-1)(2)=93 \implies n = 51  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{51} \left(2 n - 9\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B51%7D%20%5Cleft%282%20n%20-%209%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{51} \left(2 n - 9\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{51} \left(2 n - 9\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 51(-7+93) }{2}=2193 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2051%28-7%2B93%29%20%7D%7B2%7D%3D2193%20" alt="LaTeX:  \displaystyle \frac{ 51(-7+93) }{2}=2193 " data-equation-content=" \displaystyle \frac{ 51(-7+93) }{2}=2193 " /> . </p> </p>