Question

If $a={\sum }_{n=0}^{\infty }{x}^n,b={\sum }_{n=0}^{\infty }{y}^n,c={\sum }_{n=0}^{\infty }(xy{)}^n$ where $|x|,|y|<1$ ; then

• A. $abc=a+b+c$
• B. $ab+bc=ac+b$
• C. $ac+bc=ab+c$
• D. $ab+ac=bc+a$

Answer 1

The correct option is C $ac+bc=ab+c$

Clearly every summation is infinite series,
$a=\frac{1}{1-x},b=\frac{1}{1-y}$ and $c=\frac{1}{1-xy}$
or $x=1-\frac{1}{a},y=1-\frac{1}{b}$
Simplifying above equation we get, $ac+bc=ab+c$