Question

$a=2,d=8,S_n=90,$ find n.

Answer 1

$\begin{array}{c}a=2,d=8,S_n=90\\ n=?\\ weknowthat\\ S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]\\ \Rightarrow 90=\frac{n}{2}\left[4+\left(n-1\right)8\right]\\ \Rightarrow 90=\frac{n}{2}\times 2\left[2+4n-4\right]\\ \Rightarrow 90=n\left[4n-2\right]\\ \Rightarrow 45=2n^2-n\\ \Rightarrow 2n^2-n-45=0\\ \Rightarrow 2n^2-10n+9n-45=0\\ \Rightarrow 2n\left(n-5\right)+9\left(n-5\right)=0\\ \Rightarrow \left(n-5\right)\left(2n+9\right)=0\\ h=5orn=-\frac{9}{5}\left(itisnotpossible\right)\\ n=5\end{array}$