Let $a_1, a_2, a_3, \ldots$ be a sequence of real numbers. Let $s_n=a_1+a_2+\cdots+a_n$. If $2 s_n=$ $n\left(c+a_n\right)$ for some real number. for some real number $c$ and for all $n=1,2,3, \ldots$, then which one of the following statements is Correct?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a strictly decreasing function with $|f(t)|<\pi / 2$ for all $t \in \mathbf{R}$. Let $g:[0, \pi] \rightarrow$ R be a function defined by $g(t)=\sin (f(t))$. Which one of the following statements is Correct?

Let $f, g: R \rightarrow R$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?

What is the largest area of a rectangle, whose sides are parallel to the coordinate axes, that can be inscribed under the graph of the curve $y=1-x^2$ and above the $x$-axis?