if one geometric mean G and two arithmetic means p and q be inserted between two positive quantities show that - Maths - Sequences and Series

Question

if one geometric mean G and two arithmetic means p and q be
inserted between two positive quantities show that G2
=(2p-q) (2q-p)

Varun.Rawat · Accepted Answer

Let a and b be the 2 given quantities
Then G is the GM of a and b
so, G2 = ab    
1Now, p and q are 2 arithmetic means between a and b
Now, a, p, q and b form an AP with common difference, d = b-a3Now, p = a+d = a+b-a3 = 2a+b3q = a+2d = a + 2b-a3 = a+2b32p - q = 22a+b3-a+2b3 = aand 2q-p = 2a+2b3 - 2a+b3 = bNow, 2p - q2q-p = ab⇒2p - q2q-p = G2   Using 1

if one geometric mean G and two arithmetic means p and q be inserted between two positive quantities show that G2 =(2p-q) (2q-p)

if one geometric mean G and two arithmetic means p and q be inserted between two positive quantities show that G² =(2p-q) (2q-p)