You visited us 1 times! Enjoying our articles? Unlock Full Access!

Simplification of an Expression

If one AM 'a'...

Question

If one AM 'a' & two GM's p & q be inserted between any two given numbers then $p^{3} + q^{3} = k a p q$ then find $k .$

Open in App

Solution

Let the numbers be $x$ and $y$ A.M of $x$ and $y = a$

$\Rightarrow \frac{x + y}{2} = a \Rightarrow x + y = 2 a$ ...(1)

$p$ and $q$ be the G.M's between $x$ and $y$

$\Rightarrow x, p, q, y \to G . P$

$\Rightarrow p^{2} = x q$ and $q^{2} = p y$

$\Rightarrow p^{3} = p x y$ and $q^{3} = q p y$

Now

$p^{3} + q^{3} = p q x + p q y = p q (x + y) = 2 a p q$

Now comparing with $k a p q$
$∴ k = 2$

Suggest Corrections

Similar questions

A . M . A

p

q

p^{3} + q^{3} = 2 A p q

G . M .

p

q

G^{2} = (2 p - q) (2 q - p)

\frac{p^{2}}{q} + \frac{q^{2}}{p} = ?

G

p

q

G^{2} =

\frac{p^{3} + q^{3}}{p q}

\frac{p^{3} - q^{3}}{p q}

\frac{p^{2} + q^{2}}{2}

\frac{p q}{2}

Join BYJU'S Learning Program