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Write the sum \(\displaystyle 33+32+31 \ldots +13+12\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=32-(33)=-1\). Using the first term gives the sequene \(\displaystyle a_n= 33+(n-1)(-1)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 33+(n-1)(-1)=12 \implies n = 22 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{22} \left(34 - n\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 22(33+12) }{2}=495\).

Download \(\LaTeX\) 

\begin{question}Write the sum $33+32+31 \ldots +13+12$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=32-(33)=-1$. Using the first term gives the sequene $a_n= 33+(n-1)(-1)$. Setting the general term equal to the last term and solving for $n$ gives $33+(n-1)(-1)=12 \implies n = 22 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{22} \left(34 - n\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 22(33+12) }{2}=495$. }

\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 33+32+31 \ldots +13+12 " src="/equation_images/%20%5Cdisplaystyle%2033%2B32%2B31%20%5Cldots%20%2B13%2B12%20" alt="LaTeX:  \displaystyle 33+32+31 \ldots +13+12 " data-equation-content=" \displaystyle 33+32+31 \ldots +13+12 " />  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=32-(33)=-1 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D32-%2833%29%3D-1%20" alt="LaTeX:  \displaystyle a_2-a_1=32-(33)=-1 " data-equation-content=" \displaystyle a_2-a_1=32-(33)=-1 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 33+(n-1)(-1) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2033%2B%28n-1%29%28-1%29%20" alt="LaTeX:  \displaystyle a_n= 33+(n-1)(-1) " data-equation-content=" \displaystyle a_n= 33+(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 33+(n-1)(-1)=12 \implies n = 22  " src="/equation_images/%20%5Cdisplaystyle%2033%2B%28n-1%29%28-1%29%3D12%20%5Cimplies%20n%20%3D%2022%20%20" alt="LaTeX:  \displaystyle 33+(n-1)(-1)=12 \implies n = 22  " data-equation-content=" \displaystyle 33+(n-1)(-1)=12 \implies n = 22  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{22} \left(34 - n\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B22%7D%20%5Cleft%2834%20-%20n%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{22} \left(34 - n\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{22} \left(34 - n\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 22(33+12) }{2}=495 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2022%2833%2B12%29%20%7D%7B2%7D%3D495%20" alt="LaTeX:  \displaystyle \frac{ 22(33+12) }{2}=495 " data-equation-content=" \displaystyle \frac{ 22(33+12) }{2}=495 " /> . </p> </p>