Let the function f: [0, 1] → ℝ be defined by f(x) = 4 ^ x 4 ^ x + 2...

$f\left(x\right) = \dfrac{{4}^{x}}{{4}^{x} + 2}, f\left(1 - x\right) = \dfrac{{4}^{1 - x}}{{4}^{1 - x} + 2} = \dfrac{2}{{4}^{x} + 2}$

$⇒ f\left(x\right) + f\left(1 - x\right) = 1$

$f\left(\dfrac{1}{40}\right) + f\left(\dfrac{2}{40}\right) + \ldots . . + f\left(\dfrac{39}{40}\right) - f\left(\dfrac{1}{2}\right)$

$= \left(f\left(\dfrac{1}{40}\right) + f\left(\dfrac{39}{40}\right)\right) + \left(f\left(\dfrac{2}{40}\right) + f\left(\dfrac{38}{40}\right)\right) + . . . . . \left(f\left(\dfrac{19}{40}\right) + f\left(\dfrac{21}{40}\right)\right) + f\left(\dfrac{20}{40}\right) - f\left(\dfrac{1}{2}\right)$

$= 19$.