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Write the sum \(\displaystyle -29-24-19 \ldots +66+71\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=-24-(-29)=5\). Using the first term gives the sequene \(\displaystyle a_n= -29+(n-1)(5)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle -29+(n-1)(5)=71 \implies n = 21 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{21} \left(5 n - 34\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 21(-29+71) }{2}=441\).

Download \(\LaTeX\) 

\begin{question}Write the sum $-29-24-19 \ldots +66+71$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=-24-(-29)=5$. Using the first term gives the sequene $a_n= -29+(n-1)(5)$. Setting the general term equal to the last term and solving for $n$ gives $-29+(n-1)(5)=71 \implies n = 21 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{21} \left(5 n - 34\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 21(-29+71) }{2}=441$. }

\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 -29-24-19 \ldots +66+71 " src="/equation_images/%20%5Cdisplaystyle%20-29-24-19%20%5Cldots%20%2B66%2B71%20" alt="LaTeX:  \displaystyle -29-24-19 \ldots +66+71 " data-equation-content=" \displaystyle -29-24-19 \ldots +66+71 " />  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=-24-(-29)=5 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D-24-%28-29%29%3D5%20" alt="LaTeX:  \displaystyle a_2-a_1=-24-(-29)=5 " data-equation-content=" \displaystyle a_2-a_1=-24-(-29)=5 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= -29+(n-1)(5) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%20-29%2B%28n-1%29%285%29%20" alt="LaTeX:  \displaystyle a_n= -29+(n-1)(5) " data-equation-content=" \displaystyle a_n= -29+(n-1)(5) " /> . 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 -29+(n-1)(5)=71 \implies n = 21  " src="/equation_images/%20%5Cdisplaystyle%20-29%2B%28n-1%29%285%29%3D71%20%5Cimplies%20n%20%3D%2021%20%20" alt="LaTeX:  \displaystyle -29+(n-1)(5)=71 \implies n = 21  " data-equation-content=" \displaystyle -29+(n-1)(5)=71 \implies n = 21  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{21} \left(5 n - 34\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B21%7D%20%5Cleft%285%20n%20-%2034%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{21} \left(5 n - 34\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{21} \left(5 n - 34\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 21(-29+71) }{2}=441 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2021%28-29%2B71%29%20%7D%7B2%7D%3D441%20" alt="LaTeX:  \displaystyle \frac{ 21(-29+71) }{2}=441 " data-equation-content=" \displaystyle \frac{ 21(-29+71) }{2}=441 " /> . </p> </p>