Question

If three distinct real numbers a, b, c are in G.P and $a+b+c=ax$, then

• A. $xϵ[\frac{3}{4},\infty ]$
• B. $xϵR^{+}$
• C. $xϵ(-1,\infty )$
• D. none of these

Answer 1

The correct option is A $xϵ[\frac{3}{4},\infty ]$

Solution : a,b,c, in G.P $a=\frac{a_1}{r}b=a_1,c=a_{1}r$

$a+b+c=ax$

$\frac{a_1}{r}+a_1+a_{1}r=\frac{a_1}{r}x$

$a_1+a_{1}r=\frac{a_{1}r}{r}(x-1)$

$(1+r)r=x-1$

$r^2+r-x+1=0$

$D>0$

$1+4(x-1)>0$

$4x-3>0$

$x>\frac{3}{4.}$