If A is the arithmetic mean and p and q be two geometric means between two numbers a and b, then prove that :
`p^(3)+q^(3)=2pq " A"`

Answer

N/a

Explanation

The A.M. of a and b is A.
`therefore" "A=(a+b)/(2)" ...(1)"`
p and q are the G.M.'s between a and b.
`therefore" "a, p, q, b are in G.P.`
For first three numbers
`p^(2)=aq`
`rArr" "(p^(2))/(q)=a" …(2)"`
For last three numbers
`q^(2)=pb`
`rArr" "(q^(2))/(p)=b" ...(3)"`
Adding eqs. (2) and (3)
`(p^(2))/(q)+(q^(2))/(p)=a+b`
`rArr" "(p^(3)+q^(3))/(pq)=2A" [From eq. (1)]"`
`rArr" "p^(3)+q^(3)=2pqA.`