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

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

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

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

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

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

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

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

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. \(a \propto b^2\) and \(1+b \propto 6\), and if \(a = 1\) when \(c = 9\) and \(b = 5\), determine the value of \(c\).

16. Given: \[ a \propto b \quad \text{and} \quad b \propto c \] Show that: \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]