Question

If f(x) = 1 - f(1-x) and then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+.....+f\left(\frac{98}{100}\right)+f\left(\frac{99}{100}\right).$

• A. 99
• B. 49.5
• C. 49
• D. 98

Answer 1

The correct option is B

49.5

$f\left(x\right)=1-f(1-x)\Rightarrow f\left(x\right)+f(1-x)=1$ $f\left(\frac{1}{2}\right)=1-f\left(\frac{1}{2}\right)\Rightarrow f\left(\frac{1}{2}\right)=\frac{1}{2}$

$f\left(\frac{99}{100}\right)=f(1-\frac{1}{100})=1-f\left(\frac{1}{100}\right)$

$f\left(\frac{98}{100}\right)=f(1-\frac{2}{100})=1-f\left(\frac{2}{100}\right)$

So,$f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+.....+f\left(\frac{98}{100}\right)+f\left(\frac{99}{100}\right)$

$=f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+.....+1-f\left(\frac{2}{100}\right)+1-f\left(\frac{2}{100}\right)$

$\left[f\left(\frac{50}{100}\right)\right]=f\left(\frac{1}{2}\right)=\frac{1}{2}$

$=49+f\left(\frac{50}{100}\right)$

$=49+\frac{1}{2}=49.5$

Hence option (b).