Question

Let 'x' be the arithmetic mean and y,z be the two geometric means between any two positive numbers. The value of $\frac{y^3+z^3}{xyz}$ is

• A. 2
• B. 3
• C. 1/2
• D. 3/2

Answer 1

The correct option is A

2

Assumption Technique

This is a question with variables. Thus, we can assume any value which satisfies the basic conditions for x,y and z

Assume any Geometric Progession, say 1 2 4 8 .

This implies that the 2 Geometric Means between 1 and 8 are

y=2 and z=4.

Now, Arithmetic Mean, x = \(\frac {8 + 1}{2} = 4.5 .Substitutethe values for x,y,z in $\frac{y^3+z^3}{xyz}$

Answer = $\frac{2^3+4^3}{2\times 4\times 4.5}$ = 2.

Shortcut of a shortcut!

You can assume ANY Geometric Progression.

Assume the most basic GP 1 1 1 1 and the most basic AP 1 1 1,

then x=1, y=1 z=1. Answer on substitution in $\frac{y^3+z^3}{xyz}$=2, which will make the calculation even faster.

Points to note:
1.The moment you get values for x,y,z you are in a position to solve any question on x,y,z and not just the above one. Thus, all variable based questions can be solved in no time at all by assuming simple numbers!
2. You can assume any GP and solve the question. You will get the same answer.