Question
Find the mean proportional between $a-b$ and $a^3-a^2 b$
✓
Answer
Let the mean proportional between $a – b$ and $a^3 – a^2b be x.$
$\Rightarrow a – b, x, a^3 – a^2b.$ are in continued proportion.
$\Rightarrow a – b : x = x : a^3 – a^2b$
$\Rightarrow x \times x = (a – b) (a^3 – a^2b)$
$\Rightarrow x^2 = (a – b) a^2(a – b) = [a(a – b)]^2$
$\Rightarrow x = a(a – b)$
$\Rightarrow a – b, x, a^3 – a^2b.$ are in continued proportion.
$\Rightarrow a – b : x = x : a^3 – a^2b$
$\Rightarrow x \times x = (a – b) (a^3 – a^2b)$
$\Rightarrow x^2 = (a – b) a^2(a – b) = [a(a – b)]^2$
$\Rightarrow x = a(a – b)$
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