# Formulas of Sequence and Series

• The $nth$ term ${a_n}$ of the Arithmetic Progression (A.P) $a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots$is given by ${a_n} = a + (n - 1)d$.

• Arithmetic mean between $a$ and $b$ is given by $A.M = \frac{{a + b}}{2}$.

• If ${S_n}$ denotes the sum up to $n$ terms of A.P. $a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots$ then ${S_n} = \frac{n}{2}(a + l)$ where $l$ stands for last term, ${S_n} = \frac{n}{2}[2a + (n - 1)d]$

• The sum of $n$ A.M’s between $a$ and $b$ is $= \frac{{n(a + b)}}{2}$.

• The $nth$ term ${a_n}$ of the geometric progression $a,{\text{ }}ar,{\text{ }}a{r^2},{\text{ }}a{r^3}, \ldots$is ${a_n} = a{r^{n - 1}}$.

• Geometric mean between $a$ and $b$ is $G.M = \pm \sqrt {ab}$.

• If ${S_n}$ denotes the sum up to $n$terms of G.P is ${S_n} = \frac{{a(1 - {r^n})}}{{1 - r}};{\text{ }}r \ne 1$, ${S_n} = \frac{{a - rl}}{{1 - r}};{\text{ }}l = a{r^n}$ where $\left| r \right| < 1$

• The sum $S$ of infinite geometric series is $S = \frac{a}{{1 - r}};{\text{ }}\left| r \right| < 1$

• The $nth$ term ${a_n}$ of the harmonic progression is ${a_n} = \frac{1}{{a + (n - 1)d}}$.

•  Harmonic mean between $a$ and $b$ is $H.M = \frac{{2ab}}{{a + b}}$.

• ${G^2} = A \cdot H$ and $A > G > H$; where $A,G,H$ are usual notations.