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Question

If x be the arithmetic mean and y,z be two geometric means between two positive numbers, then the value of y3+z3xyz is

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Solution

Assuming the two numbers be a and b.
Now, we know that,
x=a+b2 (1)
And a, y, z, b are in G.P. so,
r=(ba)1/3y=a(ba)1/3 (2)z=a(ba)2/3 (3)

Now, using equation (1),(2) and (3),
y3+z3xyz=a3×ba+a3×b2a2ab(a+b2)=2(a2b+ab2)a2b+ab2=2

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