Given the function $f(x) = \frac{a^x + a^{-x}}{2}$,where $a > 2$. Then $f(x + y) + f(x - y) = $

If $f: R \rightarrow R$ is defined as $f(x+y)=f(x)+f(y)$,$\forall x, y \in R$ and $f(1)=5$,then find the value of $\sum_{r=1}^n f(r)$.

Let $f: R^{+} \rightarrow R^{+}$ be a function satisfying $f(x) - x = \lambda$ (constant),$\forall x \in R^{+}$ and $f(x f(y)) = f(x y) + x, \forall x, y \in R^{+}$. Then $\lim _{x \rightarrow 0} \frac{(f(x))^{\frac{1}{3}} - 1}{(f(x))^{\frac{1}{2}} - 1} =$

If $f(x) = x^{2} - 3x + 4$ and $f(x) = f(2x + 1)$,then $x =$

Let $f$ be a polynomial function such that $\log_2(f(x)) = (\log_2 (2 + \frac{2}{3} + \frac{2}{9} + \dots \infty)) \cdot \log_3 (1 + \frac{f(x)}{f(1/x)}), x > 0$ and $f(6) = 37$. Then $\sum_{n=1}^{10} f(n)$ is equal to:

If $f(x+2y, x-2y) = xy$,then $f(x, y)$ is equal to