Let f be a function with the following properties: (i) f(1) =1 (ii) f(2n)=n.f(n) for any positive integer n. Find the value of f(2^(10) ) a)1 b) 2^(10 ) c) 2^(35) d) 2^(45)


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Question Number 207085 by necx122 last updated on 06/May/24

$L e t f b e a f u n c t i o n w i t h t h e f o l l o w i n g$ $p r o p e r t i e s : (i) f (1) = 1 (i i) f (2 n) = n . f (n) f o r$ $a n y p o s i t i v e i n t e g e r n . F i n d t h e v a l u e$ $o f f (2^{10})$ $a) 1 b) 2^{10} c) 2^{35} d) 2^{45}$
	Answered by A5T last updated on 06/May/24

	$f (2 n) = n f (n)$ $f (n) = \frac{n}{2} f (\frac{n}{2}) = \frac{n}{2} \times \frac{n}{4} f (\frac{n}{4})$ $= \frac{n^{q}}{2^{1 + 2 + 3 + . . . + q}} f (\frac{n}{2^{q}}) [b y i n d u c t i o n]$ $\Rightarrow f (2^{10}) = \frac{{(2^{10})}^{10}}{2^{1 + 2 + 3 + . . . + 10}} f (\frac{2^{10}}{2^{10}}) = \frac{2^{100}}{2^{55}} = 2^{45}$
	Commented by A5T last updated on 06/May/24

	$f (2^{10}) = f (2 \times 2^{9}) = 2^{9} \times f (2^{9}) = 2^{9} \times [2^{8} \times f (2^{8})]$ $= 2^{9 + 8} \times 2^{7} f (2^{7}) = 2^{9 + 8 + 7 + . . . + 3} \times 2^{2} f (2^{2})$ $= 2^{9 + 8 + 7 + . . . + 2} \times 2 f (2) = 2^{45}$
	Commented by necx122 last updated on 06/May/24

	$T h a n k y o u s i r . I k n o w y o u t r i e d y o u r$ $b e s t t o e x p a i n y e t, I s t i l l d o n t u n d e r s t a n d$
	Commented by necx122 last updated on 06/May/24

	$T h i s n o w, i s v e r y c l e a r . T h a n k y o u s o$ $m u c h g r e a t t e a c h e r .$
	Answered by A5T last updated on 06/May/24

	$f (2^{10}) = 2^{9} f (2^{9}) = 2^{9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1} f (2) = 2^{45}$
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