Question

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

Answer 1

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

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

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

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

$\Rightarrow p^2=xq$ and $q^2=py$

$\Rightarrow p^3=pxy$ and $q^3=qpy$

Now

$p^3+q^3=pqx+pqy=pq(x+y)=2apq$

Now comparing with $kapq$
$\therefore k=2$