if a, b,c are in GP prove that a^2b^2c^2(1/ a^3+1/b^3+1/c^3) = a^3+ b^3+c^3?

Answer 1

a, b, c can be of the form dx, dx^2, dx^3. However, from inspection of the relation to be proven, d drops out, so we can dispense with it and just use x, x^2, x^3. So, we're left with:

x^2 x^4 x^6 (1/x^3 + 1/x^6 + 1/x^9) = x^3 + x^6 + x^9

Working out the left side, we see that it's

x^9 + x^6 + x^3

Now, how about that, it's true. QED

Answer 2

a,b,c are in G.P..Therefore a/b=b/c=k(say), k is the common ratio
i.e., a=bk=ck^2
b=ck ;
LHS=a^2b^2c^2(1/a^3+1/b^3+1/c^3)
= (ck^2)^2 (ck)^2 c^2 (1/(ck^2)^3+1/(ck)^3+1/c^3)
=c^6k^6 (1/c^3)(1/k^6+1/k^3+1)
=c^3k^6((k^6+k^3+1)/k^6)
=c^3(k^6+k^3+1)
=(ck^2)^3+(ck)^3+c^3
=a^3+b^3+c^3 =RHS

Answer 3

quete pasa...ups!!