If $a, b, c$ are in Harmonic Progression (HP), prove that $\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ are also in HP.

Express the repeating decimal $0.2\overline{36} = 0.2363636\dots$ as a rational fraction in lowest terms using the sum of an infinite geometric series.

For any two distinct positive real numbers $a$ and $b$, prove that: (a) $AM > GM > HM$ (b) $G^2 = AH$

Find the sum of the series to $n$ terms: $$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n(n + 1)$$