A,b,c satisfying given conditions

[utkarshakash]

Homework Statement

Let a,b,c be distinct real numbers satisfying $a^3+b^3+6abc = 8c^3$ then which of the following may be correct?

A) a,c,b are in Arithmetic Progression
B) a,c,b are in Harmonic Progression
C) a+bω-2cω^2 =0
D) a+bω^2-2cω=0

The Attempt at a Solution

I could prove the first statement but I really can't figure out how to deal with the other options. Since the first statement is correct all I know is that 2c=a+b. Suppose I replace a with 2c-b in the third statement and simplify it, I get 4c-b+ω(2c+b) = 0. But I guess LHS can't be zero(I'm not sure about this).

 

[Saitama]

Homework Statement

Let a,b,c be distinct real numbers satisfying $a^3+b^3+6abc = 8c^3$ then which of the following may be correct?

A) a,c,b are in Arithmetic Progression
B) a,c,b are in Harmonic Progression
C) a+bω-2cω^2 =0
D) a+bω^2-2cω=0

The Attempt at a Solution

I could prove the first statement but I really can't figure out how to deal with the other options. Since the first statement is correct all I know is that 2c=a+b. Suppose I replace a with 2c-b in the third statement and simplify it, I get 4c-b+ω(2c+b) = 0. But I guess LHS can't be zero(I'm not sure about this).

The following should help:
$$x^3+y^3+z^3-3xyz=(x+y+z)(x+\omega y+\omega^2 z)(x+\omega^2 y+\omega z)$$
Can you figure out what $x,y,z$ are for the given problem? :)

 

[AlephZero]

For B), do you remember an inequality between the arithmetic and geometric means of a set of numbers?

 

[Saitama]

For B), do you remember an inequality between the arithmetic and geometric means of a set of numbers?

I am not sure but OP already proved that a,b,c are in AP. Why check the B case now? :/

 

[AlephZero]

I am not sure but OP already proved that a,b,c are in AP. Why check the B case now? :/

The question asks if a b and c may be in AP, GP, etc.

The OP proved they may be in AP. There could be other values that are in GP.

In fact they could be in a GP, except that the question says they are distinct real numbers.

NOTE: I'm can't remember what I was thinking about when I posted #3 - ignore it!

You can check for a GP by substituting $b = ka$, and $c = k^2a$.

 

[Saitama]

but the question did not say they are distinct real numbers.

It does state that they are distinct. :)

 

[AlephZero]

You replied to my post #5 before I had finished editing the typos

 

[Saitama]

You replied to my post #5 before I had finished editing the typos

So no need to check the other cases now. :)