Question

suppose are the a , b c are in the A . P . $a^2,b^2,c^2$ are in G . P . If a < b < c and a + b + c = $\frac{3}{2}$then find the value of a

Answer 1

2b = a + c and a + b + c = 3/2
3b = 3 /2 or b = 1/2
Also $b^4=a^2{c}^2={\left(\frac{1}{2}\right)}^4=\frac{1}{16}\Rightarrow ac=\pm \frac{1}{4}$
Also a + c = 2b = 1
$\therefore$ a and c are the roots of
$x^2-x\pm \frac{1}{4}=0$

$\therefore x=\frac{1\pm \sqrt{1\pm 1}}{2}=\frac{1}{2},\frac{1}{2}or\frac{1}{2}\pm \frac{1}{\sqrt{2}}$