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Write the sum \(\displaystyle 43+39+35 \ldots -89-93\) in sigma notation and then find the sum.

The common difference is given by \(\displaystyle a_2-a_1=39-(43)=-4\). Using the first term gives the sequene \(\displaystyle a_n= 43+(n-1)(-4)\). Setting the general term equal to the last term and solving for \(\displaystyle n\) gives \(\displaystyle 43+(n-1)(-4)=-93 \implies n = 35 \). Writing in sigma notation gives \(\displaystyle \displaystyle \sum_{n=1}^{35} \left(47 - 4 n\right)\). Using the formula for a finite arithmetic sum gives \(\displaystyle \frac{ 35(43-93) }{2}=-875\).

Download \(\LaTeX\) 

\begin{question}Write the sum $43+39+35 \ldots -89-93$ in sigma notation and then find the sum.
    \soln{9cm}{The common difference is given by $a_2-a_1=39-(43)=-4$. Using the first term gives the sequene $a_n= 43+(n-1)(-4)$. Setting the general term equal to the last term and solving for $n$ gives $43+(n-1)(-4)=-93 \implies n = 35 $. Writing in sigma notation gives $\displaystyle \sum_{n=1}^{35} \left(47 - 4 n\right)$. Using the formula for a finite arithmetic sum gives $\frac{ 35(43-93) }{2}=-875$. }

\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 43+39+35 \ldots -89-93 " src="/equation_images/%20%5Cdisplaystyle%2043%2B39%2B35%20%5Cldots%20-89-93%20" alt="LaTeX:  \displaystyle 43+39+35 \ldots -89-93 " data-equation-content=" \displaystyle 43+39+35 \ldots -89-93 " />  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=39-(43)=-4 " src="/equation_images/%20%5Cdisplaystyle%20a_2-a_1%3D39-%2843%29%3D-4%20" alt="LaTeX:  \displaystyle a_2-a_1=39-(43)=-4 " data-equation-content=" \displaystyle a_2-a_1=39-(43)=-4 " /> . Using the first term gives the sequene  <img class="equation_image" title=" \displaystyle a_n= 43+(n-1)(-4) " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D%2043%2B%28n-1%29%28-4%29%20" alt="LaTeX:  \displaystyle a_n= 43+(n-1)(-4) " data-equation-content=" \displaystyle a_n= 43+(n-1)(-4) " /> . 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 43+(n-1)(-4)=-93 \implies n = 35  " src="/equation_images/%20%5Cdisplaystyle%2043%2B%28n-1%29%28-4%29%3D-93%20%5Cimplies%20n%20%3D%2035%20%20" alt="LaTeX:  \displaystyle 43+(n-1)(-4)=-93 \implies n = 35  " data-equation-content=" \displaystyle 43+(n-1)(-4)=-93 \implies n = 35  " /> . Writing in sigma notation gives  <img class="equation_image" title=" \displaystyle \displaystyle \sum_{n=1}^{35} \left(47 - 4 n\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%20%5Csum_%7Bn%3D1%7D%5E%7B35%7D%20%5Cleft%2847%20-%204%20n%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle \sum_{n=1}^{35} \left(47 - 4 n\right) " data-equation-content=" \displaystyle \displaystyle \sum_{n=1}^{35} \left(47 - 4 n\right) " /> . Using the formula for a finite arithmetic sum gives  <img class="equation_image" title=" \displaystyle \frac{ 35(43-93) }{2}=-875 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%2035%2843-93%29%20%7D%7B2%7D%3D-875%20" alt="LaTeX:  \displaystyle \frac{ 35(43-93) }{2}=-875 " data-equation-content=" \displaystyle \frac{ 35(43-93) }{2}=-875 " /> . </p> </p>