Question

$a,b,c$ are the first three terms of a G.P. If the harmonic mean of $a$ and $b$ is $20$ and the arithmetic mean of $b$ and $c$ is $5,$ then

• A. no term of this G.P. is the square of an integer
• B. the arithmetic mean of $a,b$ and $c$ is $5$
• C. $b=\pm 6$
• D. the common ratio of this G.P. is $2$

Answer 1

Answer: (A), (B)

The correct options are
A no term of this G.P. is the square of an integer
B the arithmetic mean of $a,b$ and $c$ is $5$

$a,b,c$ are in G.P.
Let $r$ be the common ratio. Hence, the terms can be written as $a,ar,a{r}^2$.
The harmonic mean of $a$ and $b$ is $20$.
$\Rightarrow \frac{2ab}{a+b}=20$
$\Rightarrow \frac{2a\times ar}{a(1+r)}=20$
$\Rightarrow ar=10(1+r)\to \left(1\right)$
The arithmetic mean of $b$ and $c$ is $5$.
$\Rightarrow \frac{b+c}{2}=5$
$\Rightarrow \frac{ar+a{r}^2}{2}=5$
$\Rightarrow ar(1+r)=10\to \left(2\right)$
Using $\left(1\right)$ and $\left(2\right)$, we get
$(1+r{)}^2=1$
$\Rightarrow r=0,-2$
$r$ cannot be $0$. Hence, $r=-2$
$a=5,b=-10$ and $c=20$
Any term of the G.P can be written as $5\times (-2{)}^n$.
Hence, no term of the G.P can be a perfect square.
The arithmetic mean of $a,b$ and $c=\frac{5-10+20}{3}=5$.
Hence, options A and B are correct.