Question

A function f(x) is given by f(x) = $\frac{5^x}{5^x + 5}$, then the sum of the series $f(\frac{1}{20}) + f(\frac{2}{20}) + \dots + f(\frac{39}{20})$ is equal to:

• A. $\frac{29}{2}$
• B. $\frac{49}{2}$
• C. $\frac{39}{2}$
• D. $\frac{19}{2}$

Hint:

For series of the form $\sum f(x_i)$, always check for symmetry properties of the function, such as $f(x) + f(a-x) = c$. This allows pairing terms from the beginning and end of the series to simplify the sum significantly.

Answer 1

The Correct Option is C

Evaluate: \[ f(2-x)=\frac{5^{2-x}}{5^{2-x}+5} =\frac{25/5^x}{25/5^x+5} =\frac{5}{5+5^x} \] Hence, \[ f(x)+f(2-x)=\frac{5^x+5}{5^x+5}=1 \] Pairing terms: \[ f\!\left(\frac{1}{20}\right)+f\!\left(\frac{39}{20}\right)=1 \] \[ f\!\left(\frac{2}{20}\right)+f\!\left(\frac{38}{20}\right)=1 \] There are 19 such pairs: \[ S=19+f(1) \] \[ f(1)=\frac{5}{10}=\frac{1}{2} \] \[ S=19+\frac{1}{2}=\frac{39}{2} \] Correct option: (C)