If x + 5 is the mean proportion between x + 2 and x + 9; find the value of x.

Given,

x + 5 is the mean proportion between x + 2 and x + 9

$\therefore x + 2 : x + 5 = x + 5 : x + 9 \\[1em] \Rightarrow \dfrac{x + 2}{x + 5} = \dfrac{x + 5}{x + 9} \\[1em] \Rightarrow (x + 2)(x + 9) = (x + 5)(x + 5) \\[1em] \Rightarrow x^2 + 9x + 2x + 18 = x^2 + 5x + 5x + 25 \\[1em] \Rightarrow x^2 + 11x + 18 = x^2 + 10x + 25 \\[1em] \Rightarrow 11x - 10x = 25 - 18 \\[1em] \Rightarrow x = 7.$

Hence, x = 7.