Key Points

A sequence is an ordered list of numbers, called terms, arranged according to a specific rule. The $n^{\text{th}}$ term is denoted by $a_n$.

A sequence is finite if it has a limited number of terms. It is infinite if it continues indefinitely.

A series is the sum of the terms of a sequence. The sum of the first n terms $a_1 + a_2 + \ldots + a_n$ is compactly written as $\sum_{k=1}^{n} a_k$.

A Geometric Progression (G.P.) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).

The $n^{\text{th}}$ term ($a_n$) of a G.P. with first term 'a' and common ratio 'r' is given by the formula $a_n = ar^{n-1}$.

The sum ($S_n$) of the first n terms of a G.P. is $S_n = \frac{a(r^n - 1)}{r-1}$ or $S_n = \frac{a(1 - r^n)}{1-r}$, provided $r \neq 1$. If $r=1$, then $S_n = na$.

The $n^{\text{th}}$ term ($a_n$) of an Arithmetic Progression (A.P.) with first term 'a' and common difference 'd' is given by $a_n = a + (n-1)d$.

The sum ($S_n$) of the first n terms of an A.P. is given by $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}(a+l)$, where 'l' is the last term.

The Arithmetic Mean (A.M.) of two numbers 'a' and 'b' is defined as $A = \frac{a+b}{2}$.

The Geometric Mean (G.M.) of two positive numbers 'a' and 'b' is defined as $G = \sqrt{ab}$.

For any two positive real numbers, their Arithmetic Mean is always greater than or equal to their Geometric Mean. This is expressed as $A.M. \geq G.M.$ or $\frac{a+b}{2} \geq \sqrt{ab}$.

The sum of the first 'n' natural numbers is given by the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.

The sum of the squares of the first 'n' natural numbers is given by $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.

The sum of the cubes of the first 'n' natural numbers is $\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$.

The Fibonacci sequence is defined by the recurrence relation $a_1 = 1, a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n > 2$. The sequence begins $1, 1, 2, 3, 5, 8, \ldots$.