1. **Problem (a):** Check if the proposition $$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n-1)}{2}\right]^2$$ is true for all positive integers $n$ based on the given argument. 2. The argument tests the formula for $n=1,2,3,4$ and finds the left-hand side (l.h.s.) equals the right-hand side (r.h.s.) in each case. 3. However, the formula given is incorrect. The known correct formula for the sum of cubes is: $$\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2$$ not $$\left[\frac{n(n-1)}{2}\right]^2$$. 4. The mistake is in the formula itself: the argument verifies a wrong formula for some values but does not prove it for all $n$. 5. Testing $n=1$ to $4$ is not sufficient to prove the formula for all positive integers; this is a fallacy called "proof by example". --- 6. **Problem (b):** Solve the simultaneous equations: $$2C + 3c = 10$$ $$4C - 4c = 0$$ 7. The argument claims from the first equation: $$2C + 3c = 10 \Rightarrow 5c = 10 \Rightarrow c = 2$$ 8. This step is incorrect because $2C + 3c = 10$ does not imply $5c=10$ unless $2C=0$, which is not given. 9. The mistake is treating $2C + 3c = 10$ as if $2C$ and $3c$ can be combined into $5c$; this is a fallacy. 10. Therefore, the solution $c=2$, $C=2$ is not correctly derived. --- 11. **Problem (c):** Mulingo claims that $E = f$ (elephant's mass equals fly's mass) based on the following steps: 12. Starting with: $$(E - f)^2 = [-1]^2 (E - f)^2$$ which is true since $[-1]^2 = 1$. 13. Then: $$(E - f)^2 = [-1(E - f)]^2 = (f - E)^2$$ 14. Taking square roots: $$\sqrt{(E - f)^2} = \sqrt{(f - E)^2}$$ which gives $$(E - f) = (f - E)$$ 15. This implies: $$E - f = f - E \Rightarrow 2E = 2f \Rightarrow E = f$$ 16. The fallacy is in step 14: the square root of $a^2$ is $|a|$, not $a$. 17. So, $$\sqrt{(E - f)^2} = |E - f|$$ and $$\sqrt{(f - E)^2} = |f - E| = |E - f|$$ 18. Therefore, $$|E - f| = |E - f|$$ which is always true but does not imply $E - f = f - E$. 19. The error is ignoring the absolute value and assuming the square root function returns the original expression without sign consideration. --- **Summary:** - (a) Fallacy: Incorrect formula and proof by example. - (b) Fallacy: Incorrect algebraic manipulation. - (c) Fallacy: Misuse of square root properties ignoring absolute values.