Question

What is the sum of the first 27 terms of the arithmetic sequence?
-15,-11,-7,-3,..

Answer 1

`\bf -15~~,~~\stackrel{-15+4}{-11}~~,~~\stackrel{-11+4}{-7}~~,~~\stackrel{-7+4}{-3}~~~~,...\qquad \qquad \boxed{d=4} \\\\[-0.35em] ~\dotfill\\\\ n^(th)\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\[-0.5em] \hrulefill\\ d=4\\ a_1=-15\\ n=27 \end{cases} \\\\\\ a_(27)=-15+(27-1)4\implies a_(27)=-15+(26)4 \\\\\\ a_(27)=-15+104 \implies a_(27)=89`

`\bf \rule{34em}{0.25pt}\\\\ \textit{ sum of a finite arithmetic sequence} \\\\ S_n=\cfrac{n(a_1+a_n)}{2}\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\[-0.5em] \hrulefill\\ n=27\\ a_1=-15\\ a_(27)=89 \end{cases} \\\\\\ S_(27)=\cfrac{27(a_1+a_(27))}{2}\implies S_(27)=\cfrac{27(-15+89)}{2}\implies S_(27)=\cfrac{27(74)}{2} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill S_(27)=999~\hfill`