Class 11

Math

Algebra

Sequences and Series

If a, b, c are in AP; x is the GM between a and b; y is the GM between b and c; then show that $b_{2}$ is the AM between $x_{2}$ and $y_{2}$.

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

$∴(a+c)=2b,$

Also, $x$ is the GM between $a$ and $b;y$ is the GM between $b$ and $c$;

$x=ab $ and $y=bc $

$⇒x_{2}+y_{2}=(ab+bc)=b(a+c)=2b_{2}$.

$⟹b_{2}$ is the AM between $x_{2}$ and $y_{2}$.