Solve the equation for x : \[1+4+7+10+\ldots .+x=287 \]

\[\begin{aligned} & 1+4+7+10+\ldots \ldots+x=287 \\ & a=1, d=3 \text { Last term }=x \\ & \Rightarrow 1+(n-1) 3=x \\ & 3 n-2=x \Rightarrow n=\frac{x+2}{3}-(i) \\ & S_{n}=287 \\ & \frac{n}{2}[1+x]=287 \Rightarrow \frac{(x+2)(x+1)}{6}=287-\text { using (i) } \\ & x^{2}+3 x+2=1722 \\ & x^{2}+3 x-1720=0 \\ & \Rightarrow(x+43)(x-40)=0 \\ & \Rightarrow x=-43,40 \\ & x \neq-43 \\ & \therefore x=40 \end{aligned} \]