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Write the sum \(\displaystyle -27-36-45 \ldots -333-342\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=-36-(-27)=-9\). Using the first term gives the sequene \(\displaystyle a_n= -27+(n-1)(-9)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle -27+(n-1)(-9)=-342 \implies n = 36 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 36(-27-342) }{2}=-6642\).

Download \(\LaTeX\) 

\begin{question}Write the sum $-27-36-45 \ldots -333-342$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=-36-(-27)=-9$. Using the first term gives the sequene $a_n= -27+(n-1)(-9)$. Setting the general term equal to the last term and solving for $n$ gives $-27+(n-1)(-9)=-342 \implies n = 36 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 36(-27-342) }{2}=-6642$. }

\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 -27-36-45 \ldots -333-342 " src="/equation_images/%20%5Cdisplaystyle%20-27-36-45%20%5Cldots%20-333-342%20" alt="LaTeX:  \displaystyle -27-36-45 \ldots -333-342 " data-equation-content=" \displaystyle -27-36-45 \ldots -333-342 " />  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=-36-(-27)=-9 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D-36-%28-27%29%3D-9%20" alt="LaTeX:  \displaystyle a_2-a_1=-36-(-27)=-9 " data-equation-content=" \displaystyle a_2-a_1=-36-(-27)=-9 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= -27+(n-1)(-9) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%20-27%2B%28n-1%29%28-9%29%20" alt="LaTeX:  \displaystyle a_n= -27+(n-1)(-9) " data-equation-content=" \displaystyle a_n= -27+(n-1)(-9) " /> . 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 -27+(n-1)(-9)=-342 \implies n = 36  " src="/equation_images/%20%5Cdisplaystyle%20-27%2B%28n-1%29%28-9%29%3D-342%20%5Cimplies%20n%20%3D%2036%20%20" alt="LaTeX:  \displaystyle -27+(n-1)(-9)=-342 \implies n = 36  " data-equation-content=" \displaystyle -27+(n-1)(-9)=-342 \implies n = 36  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B36%7D%20%5Cleft%28-%209%20n%20-%2018%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 36(-27-342) }{2}=-6642 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2036%28-27-342%29%20%7D%7B2%7D%3D-6642%20" alt="LaTeX:  \displaystyle \frac{ 36(-27-342) }{2}=-6642 " data-equation-content=" \displaystyle \frac{ 36(-27-342) }{2}=-6642 " /> . </p> </p>