First slide

Functions (XII)

Question

$Consider a real-valued function f (x) satisfying 2 f (x y)$ $= (f (x))^{y} + (f (y)^{x} \forall x, y \in R and f (1) = a where a \neq 1 . then$ $(a - 1) \sum_{l = 1}^{n} f (i) =$

Moderate

A

$a^{n} - 1$
B

$a^{n + 1} + 1$
C

$a^{n} + 1$
D

$a^{n + 1} - a$

Solution

$We have 2 f (x y) = (f (x))^{y} + (f (y))^{x}$
$Replacing y by 1, we get 2 f (x) = f (x) + (f (1))^{x} \Rightarrow f (x) = a^{x}$
$\begin{aligned} \Rightarrow \sum_{i = 1}^{n} f (i) = a + a^{2} + \dots + a^{n} = \frac{a^{n + 1} - a}{a - 1} \\ \Rightarrow (a - 1) \sum_{i = 1}^{n} f (i) = a^{n + 1} - a \end{aligned}$

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