Question

If $|a|,|b|,|c|<1$ and $a,b,cϵA.P$. then $(1+a+a^2+...\infty ),(1+b+b^2+...\infty ),(1+c+c^2+...\infty )$ are in

• A. $A.P$
• B. $G.P$
• C. $H.P$
• D. None of these

Answer 1

The correct option is C $H.P$

Consider the terms $(1+a+a^2+...\infty ),(1+b+b^2+...\infty ),(1+c+c^2+...\infty )$

Using the property for sum of an infinite G.P., we get
$1+a+a^2+a^3+...\infty =\frac{1}{1-a}$
$1+b+b^2+b^3+...\infty =\frac{1}{1-b}$
and $1+c+c^2+....\infty =\frac{1}{1-c}$
Given that, $a,b,c$ are in A.P.
$\Rightarrow 1-a,1-b,1-c$ are in A.P
$\Rightarrow \frac{1}{1-a},\frac{1}{1-b},\frac{1}{1-c}$ are in H.P
Therefore, $(1+a+a^2+...\infty ),(1+b+b^2+...\infty ),(1+c+c^2+...\infty )$ are in H.P.
Ans: C