Functional Problem

Calculus Level pending

Let $f$ be a function with the following properties:

$(i)\quad f(1) = 1$ , and

$(ii)\quad f(2n) = n\times f(n)$ , for any positive integer $n$ .

What is the value of $f(2^{100})$ ?

2^{7850}

2^{4950}

2^{99}

2^{100}

1

1 solution

Danish Ahmed
Mar 26, 2015

$f(2^{100})$

$= f(2 \times 2^{99})$

$= 2^{99} \times f(2^{99})$

$= 2^{99} \cdot 2^{98} \times f(2^{98})$

$= \ldots= 2^{99}2^{98}\cdots 2^{1} \cdot 1 \cdot f(1)$

$= 2^{99 + 98 + \ldots + 2 + 1}$

$= 2^{\frac{99(100)}{2}} = 2^{4950}$