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Question

Let x be the arithmetic mean and y,z be the two geometric means between any two positive numbers. Then y3+z3xyz = __________.

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Solution

Given that x is the A.M. between a and b and y, and z are the G.M. between a and b where a and b are positive. Then a,x,b are in A.P. So,
x=a+b2
a,y,z,b are in G.P. So,
y=ar and z=ar2, b=zrb=ar3
r=(ba)13
Also, yz=ab,


Now, y3+z3xyz=a3r3+a3r6ab(a+b2)

=a3×ba+a3×b2a2ab(a+b2)

=2(a2b+ab2)a2b+ab2

y3+z3xyz=2


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