Find the arithmetic mean of :

(i) -5 and 41

(ii) 3x - 2y and 3x + 2y

(iii) (m + n)² and (m - n)²

(i) Arithmetic mean = $\dfrac{-5 + 41}{2} = \dfrac{36}{2}$ = 18.

Hence, arithmetic mean between -5 and 41 = 18.

(ii) Arithmetic mean = $\dfrac{3x - 2y + 3x + 2y}{2} = \dfrac{6x}{2}$ = 3x.

Hence, arithmetic mean between 3x - 2y and 3x + 2y = 3x.

(iii)

Arithmetic mean = $\dfrac{(m + n)^2 + (m - n)^2}{2}$

$\phantom{Arithmetic mean }= \dfrac{m^2+ n^2 + 2mn + m^2 + n^2 - 2mn}{2} \\[1em] \phantom{Arithmetic mean }= \dfrac{2(m^2 + n^2)}{2} \\[1em] \phantom{Arithmetic mean }= m^2 + n^2.$.

Hence, arithmetic mean between (m + n)² and (m - n)² = m² + n².