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:

Let $f: \mathbb{R} - \{0\} \rightarrow (-\infty, 1)$ be a function satisfying $f(x)f(\frac{1}{x}) = f(x) + f(\frac{1}{x})$. If $f(x)$ is a polynomial of degree $2$ and $f(K) = -2K$,then the sum of squares of all possible values of $K$ is:

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

Let $f: R \rightarrow R$ be defined as $f(x+y)+f(x-y)=2 f(x) f(y)$ and $f\left(\frac{1}{2}\right)=-1$. Then,the value of $\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}$ is equal to:

Let $f: R \rightarrow R$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$ for all $x, y \in R$. If $f'(0) = 2$,then $|f(-2)|$ is equal to:

Suppose that a function $f: R \rightarrow R$ satisfies $f(x+y)=f(x) f(y)$ for all $x, y \in R$ and $f(1)=3$. If $\sum_{i=1}^{n} f(i)=363$,then $n$ is equal to