{ (( f : [ 0 , 1 ] → R)),(( f (x ) = (( 4^( x) )/(2 + 4^( x) )))) :} is given . find the value of the following expression. E = f ((1/(20)) )+ f((( 2)/(20)) )+... +f (((19)/(20))) −f ((1/2) )=?


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Question Number 185983 by mnjuly1970 last updated on 30/Jan/23
${ (( f : [ 0 , 1 ] → R)),(( f (x ) = (( 4^( x) )/(2 + 4^( x) )))) :} is given . find the value of the following expression. E = f ((1/(20)) )+ f((( 2)/(20)) )+... +f (((19)/(20))) −f ((1/2) )=?$
${\begin{cases} f : [0, 1] \to R \\ f (x) = \frac{4^{x}}{2 + 4^{x}} \end{cases}$ $i s g i v e n . f i n d t h e v a l u e o f$ $t h e f o l l o w i n g e x p r e s s i o n .$ $E = f (\frac{1}{20}) + f (\frac{2}{20}) + . . . + f (\frac{19}{20}) - f (\frac{1}{2}) = ?$
	Answered by ARUNG_Brandon_MBU last updated on 30/Jan/23

	$f (x) = \frac{4^{x}}{2 + 4^{x}} \Rightarrow f (1 - x) = \frac{4^{1 - x}}{2 + 4^{1 - x}} = \frac{2}{4^{x} + 2}$ $f (x) + f (1 - x) = \frac{4^{x}}{2 + 4^{x}} + \frac{2}{2 + 4^{x}} = 1$ $\Rightarrow E = \underset{k = 1}{\sum^{9}} (f (\frac{k}{20}) + f (1 - \frac{k}{20})) + f (\frac{10}{20}) - f (\frac{1}{2})$ $= \underset{k = 1}{\sum^{9}} (1) + f (\frac{1}{2}) - f (\frac{1}{2}) = 9$
	Commented by ARUNG_Brandon_MBU last updated on 30/Jan/23

	$Generally if f (x) = \frac{a^{2 x}}{a + a^{2 x}}, then f (x) + f (1 - x) = 1$
	Commented by mnjuly1970 last updated on 30/Jan/23

	$t h a n k s a l o t s i r A r u n g$
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