1. ## AP prove

if x,y,z are in AP and A1 is the AM between x and y and A2 is the AM between y and z the prove that the AM between A1 and A2

is y.

my incompelte soln-
A1 = (x + y)/2
A2 = (y + z)/2

2. Originally Posted by saha.subham
if x,y,z are in AP and A1 is the AM between x and y and A2 is the AM between y and z the prove that the AM between A1 and A2

is y.

my incompelte soln-
A1 = (x + y)/2
A2 = (y + z)/2
\displaystyle \begin{align*}A_M &= \frac{A_1 + A_2}{2} \\ &= \frac{\frac{x + y}{2} + \frac{y + z}{2}}{2} \\ &= \frac{\frac{x + 2y + z}{2}}{2} \\ &= \frac{x + 2y + z}{4} \\ &= \frac{x + z}{4} + \frac{2y}{4} \\ &= \frac{1}{2}\left(\frac{x + z}{2}\right) + \frac{y}{2} \\ &= \frac{y}{2} + \frac{y}{2} \\ &= y\end{align*}