Question

Let $a,b,c$ are in $G.P.$ and $4a,5b,4c$ are in A.p. such that $a+b+c=70$, then value of $b$ is

• A. $5,10,15$
• B. $10,20,40$
• C. $15,30,45$
• D. $20,40,60$

Answer 1

The correct option is B $10,20,40$

Given a, b, c are in G.P

Let them(the three numbers) be $a,ar,a{r}^2$

Given sum $=70$

$\Rightarrow a(1+r+r^2)=70$ ………$\left(1\right)$

also given $4a,5ar,4a{r}^2$ are in A.P

$2\left[5ar\right]=4a+4a{r}^2$ $\Rightarrow$ [If a, b, c are in A.P, $2b=a+c$]

$5r=2+2r^2$

$2r^2-5r+2=0$

$\Rightarrow r=2$ or $\frac{1}{2}$

From $\left(1\right)$ $a(1+2+2^2)=70$

$r=2$ $a\left(7\right)=70\Rightarrow a=10$

numbers $10,20,40$

$r=\frac{1}{2}$ $a(1+\frac{1}{2}+\frac{1}{2^2})=70$

$a\left(\frac{7}{4}\right)=70$

$a=40$ numbers $40,20,10$

Hence the numbers are $10,20,40$.