Question

If $a,\frac{1}{b},c$ and $\frac{1}{p},q,\frac{1}{r}$ form two A.P.'s, then $a,q,c$ are in $A.P.$ if

• A. $p,b,r$ are in $A.P.$
• B. $\frac{1}{p},\frac{1}{b},\frac{1}{r}$ are in $A.P.$
• C. Both of $p,b,r$ and $\frac{1}{p},\frac{1}{b},\frac{1}{r}$ are in $A.P.$
• D. None of above

Answer 1

The correct option is B $\frac{1}{p},\frac{1}{b},\frac{1}{r}$ are in $A.P.$

As $a,\frac{1}{b},c$ are in A.P.
$\therefore \frac{2}{b}=a+c...\left(1\right)$

Let $a,q,c$ are in $A.P.$ then
$2q=a+c...\left(2\right)$

Given $\frac{1}{p},q,\frac{1}{r}$ are also in $A.P.$, then
$\frac{1}{p}+\frac{1}{r}=2q$
Using equation $\left(1\right)$, we get
$\Rightarrow \frac{1}{p}+\frac{1}{r}=a+c$
Using equation $\left(2\right)$, we get
$\Rightarrow \frac{1}{p}+\frac{1}{r}=\frac{2}{b}$

Hence $\frac{1}{p},\frac{1}{b},\frac{1}{r}$ are in $A.P.$