Question

Suppose $a,b,c$ are in $A.P.$ and $a^2,b^2,c^2$ are in G.O. if $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is

• A. $\frac{1}{2}$
• B. $\frac{5}{2}$
• C. $\frac{3}{2}$
• D. $-\frac{1}{2}$

Answer 1

Answer: (A)

$\frac{1}{2}$

Given that,

$a,b,c$ are in $A.P.$

So, $2b=a+c$ ......(1)

$a^2,b^2,c^2$ are in $G.P.$

So, $b^4=a^2{c}^2\Rightarrow b^2=ac$ ......(2)

Since, $a+b+c=\frac{3}{2}$ ......(3)

$2b+b=\frac{3}{2}$ (from equation $\left(1\right)$)

$b=\frac{1}{2}$

On substituting this value in equation (1) and (2) respectively, we get

$a+c=1$ ......(4)

$c=\frac{1}{4a}$ ......(5)

On solving equation (4) and (5), we get

$4a^2-4a+1=0$

$4a^2-2a-2a+1=0$

$2a(2a-1)-1(2a-1)=0$

$(2a-1{)}^2=0$

$a=\frac{1}{2}$

Hence, this is the answer.