if f(x) = 16^x/(16^x + 4)
find f(1/2017) + f(2/2017) +f(3/2017) + ...+ f(2016/2017)
I have a sneaky suspicion that your function is
f(x) = 16^x/(16^(x+4) ) which would simply be
f(x) = 1/16^4 or 1/65536
in that case, f(4) = 1/65636 , f(13.56) = 1/65636 etc , doesn't matter what you have for the x value, since x does not show up in your function definition.
so f(1/2017) + f(2/2017) +f(3/2017) + ...+ f(2016/2017)
= 1/65636 + 1/65636 + 1/65636 + .... + 1/65636 , for 2016 terms
= 2016/65636
= 504/16409
If you meant it the way you typed it, we would have a rather cumbersome arithmetic problem.
should be:
f(1/2017) + f(2/2017) +f(3/2017) + ...+ f(2016/2017)
= 1/65636 + 2/65636 + 3/65636 + .... + 2016/65636 , for 2016 terms
= (1/65636)(1+2+3+...+2016)
= (1/65636)(1008)(1+2016) = 2033136/65636 = 508284/16409
the dangers of a simple typo, and the to copy-and-paste it ...
whenever you see 1/65636 it should say 1/65536>
so the final fraction is 2033136/65536 = 127071/4096
To find the value of f(1/2017) + f(2/2017) + f(3/2017) + ... + f(2016/2017), we need to substitute each of the given values into the function f(x) = 16^x / (16^x + 4) and add up the results.
Let's break down the process step by step:
1. Start with the given function f(x) = 16^x / (16^x + 4).
2. Substitute x = 1/2017 into the function:
f(1/2017) = 16^(1/2017) / (16^(1/2017) + 4)
3. Substitute x = 2/2017 into the function:
f(2/2017) = 16^(2/2017) / (16^(2/2017) + 4)
4. Continue with this process until you reach f(2016/2017).
5. Add up all the values obtained in step 3. The sum will give you the required result:
f(1/2017) + f(2/2017) + f(3/2017) + ... + f(2016/2017)
Note that actually calculating these individual values might require a calculator or a computer program, as the exponentiation and addition can become quite complex.