Question

If $a,b,c\in R^{+}$ are such that $2a,b,4c$ are in A.P. and $c,a,\text{and}b$ are in G.P., then

• A. $a^2,ac\text{and}{c}^2\text{are in A.P.}$
• B. $c,a\text{and}a+2c\text{are in A.P.}$
• C. $c,a\text{and}a+2c\text{are in G.P.}$
• D. $\frac{a}{2},c\text{and}c-a\text{are in G.P.}$

Answer 1

The correct option is C $c,a\text{and}a+2c\text{are in G.P.}$

Since, $2a,b,4c$ are in A.P.
$\therefore b=a+2c....\left[1\right]$
Also, $c,a,\text{and}b$ are in G.P.
$\therefore a^2=bc\Rightarrow a^2=(a+2c)c\text{(From [1])}$
$\therefore c,a\text{and}a+2c\text{are in G.P.}$
$a^2=ac+2c^2\Rightarrow a^2-ac=2c^2$
$\text{So,}{a}^2,ac\text{and}{c}^2\text{are not in A.P.}$
$\frac{a}{2}(a-c)=c^2$
$\text{So,}\frac{a}{2},c\text{and}c-a\text{are not in G.P.}$