The arithmetic mean of 1/(a+b) and 1/(b+c) is 1/(a+c)
Find in terms of b , the arithmetic mean of a^2 and c^2 .
Arithmetic mean implies
$\displaystyle \frac{\frac{1}{a+b} + \frac{1}{b+c}}{2} = \frac{1}{a+c}$
In this equation find a common denominator for the LHS, cross multiply then simplify.
After some cancelling you will be left with $\displaystyle a^2+c^2 = \dots $
Then divide both sides by 2 and your answer will appear.
Spoiler: