1. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\).

2. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\). Let me know if you'd like this proven step by step. I'm here for it.

3. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\).

4. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\)

5. If \(a \propto b\), and \(b \propto c\), then show that \(a^3 b^3 + b^3 c^3 + c^3 a^3 \propto abc(a^3 + b^3 + c^3)\).

6. Given that \(a \propto b\) and \(b \propto c\), show that \(a^3 + b^3 + c^3 \propto 5abc\).

7. If \(a \propto b\) and \(b \propto c\), then show that \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

8. If \(a \propto b\) and \(b \propto c\), then show that \[a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3)\]

9. Given: \(a \propto b\) and \(b \propto c\) Show that: \(a^3 + b^3 + c^3 \propto abc\)

10. If \(a \propto b\) and \(b \propto c\), then prove that \(a + b + c \propto 5abc\).

11. If \(x ∝ y\) and \(y ∝ z\), then prove that \((x^2 + y^2 + z^2) ∝ (xy + yz + zx)\).

12. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

13. If \(a\propto b\) and \(b\propto c\), prove that: \[ a^2+b^2+c^2 \propto ab+bc+ca \]

14. If the roots of \((a^2-bc)x^2+2(b^2-ca)x+(c^2-ab)=0\) are equal, show that \(b=0\) or \(a^3+b^3+c^3=3abc\).

15. If \(a \propto b+c\), \(b \propto c+a\), and \(c \propto a+b\), and \(l, m, n\) are three distinct constants, prove that \(\cfrac{l}{l+1}+\cfrac{m}{m+1}+\cfrac{n}{n+1}=1\).

16. If \(a:b = b:c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

17. If \(a\propto b\) and \(b\propto c\), then show that \((pa+qb+rc) \propto (l\sqrt{bc}+m\sqrt{ca}+n\sqrt{ab})\).

18. If \(a : b = b : c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3} + \cfrac{1}{b^3} + \cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]