Question

Find the fourth proportional to $$(a^2 - ab + b^2)$$, $$(a^3 + b^3)$$ and $$(a - b)$$.

Answer 1

Let the fourth proportional be $$x$$.

Then, $$\text{Product of extremes} = \text{Product of means}$$.

$$(a^2 - ab + b^2) : (a^3 + b^3) :: (a - b) : x$$

$$(a^2 - ab + b^2) \times x = (a^3 + b^3)(a - b)$$

Since $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$:

$$(a^2 - ab + b^2) \times x = (a + b)(a^2 - ab + b^2)(a - b)$$

$$x = (a + b)(a - b) = a^2 - b^2$$