Question

If 1 + 4 + 7 + 10 + ... + x = 287, find the value of x. [CBSE 2017]

Answer 1

$$\begin{gathered} s_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right] \\ a=1,d=4-1=3\mathrm{and}{s}_n=287 \\ 287=\frac{n}{2}\left[2+\left(n-1\right)3\right] \\ \Rightarrow 574=n\left(2+3n-3\right) \\ \Rightarrow 3n^2-n-574=0 \\ \mathrm{Using}n=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \end{gathered}$$

$$\begin{gathered} n=\frac{-\left(-1\right)\pm \sqrt{{\left(-1\right)}^2-4\left(3\right)\left(-574\right)}}{2\left(3\right)} \\ \Rightarrow n=\frac{1\pm \sqrt{1+6888}}{6} \\ \Rightarrow n=14,\frac{-41}{3} \\ \mathrm{Number}\mathrm{of}\mathrm{terms}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{negative} \\ \mathrm{So},n=14 \end{gathered}$$
$$\begin{gathered} n=14 \\ {s}_n=\frac{n}{2}\left[a+l\right] \\ 287=\frac{14}{2}\left[1+x\right] \\ \Rightarrow x=40 \end{gathered}$$