Arithmetic Progression (AP):

nth term: a_n = a_1 + (n-1)d

Sum: S_n = \frac{n}{2}[2a_1 + (n-1)d]

Geometric Progression (GP):

nth term: a_n = a_1 r^{n-1}

Sum: S_n = \frac{a_1(1-r^n)}{1-r}, r \neq 1

Harmonic Progression (HP):

Reciprocal of AP

If a, b, c are in HP, then \frac{1}{a}, \frac{1}{b}, \frac{1}{c} are in AP

Arithmetic Mean (AM):

For a, b: AM = \frac{a+b}{2}

For n numbers: AM = \frac{\sum x_i}{n}

Geometric Mean (GM):

For a, b: GM = \sqrt{ab}

For n numbers: GM = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}

Special Sums:

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}