Question

If x, y, z are in A.P. and A₁ is the A.M. of x and y and A₂ is the A.M. of y and z, then prove that the A.M. of A₁ and A₂ is y.

Answer 1

x, y, z are in A.P.

\[\therefore\] \[y = \frac{x + z}{2}\]

Now,

\[A_1\text {  is the arithmetic mean of x and y } .\]

\[A_1 = \frac{x + y}{2} = \frac{x + \frac{x + z}{2}}{2} = \frac{3x + z}{4}\]

And,

\[A_2 \text { is the arithmetic mean of y and z } .\]

\[A_2 = \frac{y + z}{2} = \frac{\frac{x + z}{2} + z}{2} = \frac{3z + x}{4}\]

Let \[A_3\] be the arithmetic mean of \[A_1 \text { and } A_2\].

\[A_3 = \frac{A_1 + A_2}{2}\]

\[ = \frac{\frac{3x + z}{4} + \frac{3z + x}{4}}{2}\]

\[ = \frac{4x + 4z}{8}\]

\[ = \frac{x + z}{2}\]

\[ = y\]

Hence, proved.