If $f(x - y) + f(x + y) = 2f(x)f(y)$ for all $x, y \in R$, then $f(x)$ is .....

If $f: R \rightarrow R$ is defined by $f(x+y)=f(x)+f(y)$ for all $x, y \in R$ and $f(1)=7$,then $\sum_{r=1}^n f(r)=$

Let $f(x)$ be a differentiable function which satisfies the equation $f(xy) = f(x) + f(y)$ for all $x > 0, y > 0$. Then $f'(x)$ is equal to:

Suppose $f : R \rightarrow (0, \infty)$ be a differentiable function such that $5f(x + y) = f(x) \cdot f(y), \forall x, y \in R$. If $f(3) = 320$,then $\sum_{n=0}^5 f(n)$ is equal to:

If $f(x)$ is a polynomial function satisfying $f(x) \cdot f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ and $f(4)=257$,then $f(3)=$

If $f : R \to R$ is such that $f(x + y) = f(x) + f(y)$ for all $x, y \in R$,$f(1) = 7$ and $\sum_{r=1}^{n} f(r) = 14112$,then $n$ is equal to: