Final answer:
The question calculates the sum of function values using a given function's property, where the sum of f(x) at various points minus the value of f(1/2) equals 19.5.
Step-by-step explanation:
The question presents a function f(x) = \frac{4^x}{4^x + 2} and asks to compute the sum of its values at points \frac{1}{40}, \frac{2}{40}, ..., \frac{39}{40}, and then subtract the value at \frac{1}{2}. This appears to be an exercise in series and function properties, likely involving symmetry or some form of function transformation.
However, upon examining the function, we see that there is potential for a symmetry property: f(x) + f(1-x) = 1. This is due to the exponents of the numerator and the denominator being mirrors of each other around \frac{1}{2}. The summation from \frac{1}{40} to \frac{39}{40} exploits this symmetry, with each pair f(\frac{n}{40}) + f(\frac{40-n}{40}) equalling 1.
Considering this, we can pair off terms in the sum until only f(\frac{1}{2}) doesn't have a pair. Since f(\frac{1}{2}) itself is equal to 1/2, subtracting it from the series, where every pair sums to 1, results in a sum of \frac{19.5}{2}.