Question
Find the third proportional to $a - b$ and $a^2-b^2$
✓
Answer
Let the third proportional to $a - b$ and $a^2-b^2$ be $x$
$\Rightarrow a - b , a ^2- b ^2, x$ are in continued proportion.
$\Rightarrow a-b: a^2-b^2=a^2-b^2: x$
$\Rightarrow \frac{a-b}{a^2-b^2}=\frac{a^2-b^2}{x} $
$ \Rightarrow x=\frac{\left(a^2-b^2\right)^2}{a-b} $
$ \Rightarrow x=\frac{(a+b)(a-b)\left(a^2-b^2\right)}{a-b} $
$ \Rightarrow x=(a+b)\left(a^2-b^2\right)$
$\Rightarrow a - b , a ^2- b ^2, x$ are in continued proportion.
$\Rightarrow a-b: a^2-b^2=a^2-b^2: x$
$\Rightarrow \frac{a-b}{a^2-b^2}=\frac{a^2-b^2}{x} $
$ \Rightarrow x=\frac{\left(a^2-b^2\right)^2}{a-b} $
$ \Rightarrow x=\frac{(a+b)(a-b)\left(a^2-b^2\right)}{a-b} $
$ \Rightarrow x=(a+b)\left(a^2-b^2\right)$
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