Question

if $f\left(x\right)=\frac{9^x}{9^x+3}$ then $f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)+.........$

• A. $997$
• B. $997.5$`
• C. $998$
• D. $998.5$

Answer 1

Answer: (B)

$997.5$`

Observing that the terms to be summed are of the form $f\left(x\right)$from one end

Putting $x$ as $(1-x)$ the other

Putting $f(1-x)$ we get,

$f(1-x)=\frac{3}{3+9^x}$

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

Coupling the terms

$f\left(\frac{1}{1996}\right)+f\left(\frac{1995}{1996}\right)=1$

Similarly $f\left(\frac{2}{1996}\right)+f\left(\frac{1994}{1996}\right)=1$

Now, $997$ such couples exist. Which gives a sum 997.

and remaining term, $f\left(\frac{998}{1996}\right)=f\left(\frac{1}{2}\right)$, Solving $=\frac{1}{2}$

thus Sum $=997+0.5=997.5$