Fill in the blanks -

If f(x)=9x/9x+3 thenf(1/1996)+f(2/1996)+...f(1995/1996) is ____.

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Detailed Solution

f(x)=9x/9x + 3
f(1−x)=91-x/91-x + 3
adding both we get, f(x) + f(1-x) = 9x +3/9x + 3 = 1
f(x)+f(1−x)=1...(a)f(x)+f(1−x)=1...(a)
Now consider,
f(1/1996)+f(2/1996)+...f(1995/1996)
The first and last term are of the form in equation (a) .
Therefore, we can group it in the form f(1/1996)+f(19951996)+f(21996)+f(19941996)+...f(11996)+f(19951996)+f(21996)+f(19941996)+...
i.e. f(1/1996)+f(1995/1996)+f(2/1996)+f(1994/1996)+...
i.e. 1+1+....
We need to find how many such pairs will be there. For that we take 1995
and divide it by 2 .
So there will be 997 such pairs and one pair will be left alone.
i.e. 1+1+....+1+f(998/1996)
= 997+f(998/1996)
Dividing 998/1996 , we get
997+f(1/2)
Now putting x = ½ in f(x) , we get f(x) = 0.5
Thus the desired result is 997.5

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