Question

If $a,b,c$ are in A.P. and $p$ is the A.M. between $a$ and $b$, and $q$is the A.M. between $b$ and $c$, show that $b$ is the arithmetic mean between $p$ and $q$.

Answer 1

Proving $b$ is the A.M. between $p$ and $q$.

Given: $a,b,c$ are in AP.

$\Rightarrow 2b=a+c$…………………… $1$

$p$ is the AM between $a$ and $b$.

$\Rightarrow$ $p=\frac{a+b}{2}$

$\Rightarrow$ $2p=a+b$…………………… $2$

Similarly $q$ is the A.M. between $b$ and $c$.

$\Rightarrow$ $q=\frac{b+c}{2}$

$\Rightarrow$ $2q=b+c$…………………… $3$

Adding equation $2$, equation $3$

$\Rightarrow$$2p+2q=b+c+a+b$

$\Rightarrow 2(p+q)=2b+(a+c)$

From equation $1$:

$\Rightarrow$$2(p+q)=2b+2b$

$\Rightarrow$$2(p+q)=4b$

$\Rightarrow$$\frac{(p+q)}{2}=b$

Hence, $b$ is the A.M. between $p$ and $q$.