# Sequences & Series

 
Sequences & Series cover Arithmetic, Geometric and Harmonic Progressions. This post will enumerate the essential formulae with a link to problems and solutions thereof. Study Complex Numbers.

## What is a Sequence in mathematics?

A sequence is an ordered collection of numbers $$t_1, t_2, t_3, … , t_n, …$$. The word of importance is ‘ordered’. It signifies a recognizable definite rule of relationship between the terms or elements. Following the rule, the general term $$t_n$$ is determinable for any value of the natural number $$n$$. A sequence can be finite or infinite.

## Arithmetic Progression

If the difference between any term and the preceding term of a sequence is constant, it is in arithmetic progression. Symbolically, $t_n – t_n-1 = k$

General Form $a + (a + d) + (a + 2d) + (a + 3d) + ⋯ +\{a + (n – 1)d\}$
How to find out a particular term? $t_n = a + (n – 1)d$
Sum of all the terms or up to a particular point $S_n = \frac{n}{2}(a + l) = \frac{n}{2}\{2a + (n – 1)d\}$
Find the Last Term itself $l = a + (n – 1)d$
You are familiar with Arithmetic Mean since long $\frac{a + b}{2}$
So, let’s insert ‘n’ A.M’s between ‘a’ and ‘b’: $\frac{an + b}{n + 1}, \frac{a(n – 1) + 2b}{n + 1}, \frac{a(n – 2) + 3b}{n + 1}, … , \frac{a + nb}{n + 1}$
Can you find sum of the ‘n’ A.M’s? Yes. $A_1 + A_2 + A_3 + ⋯ +A_n = \frac{a + b}{2}.n$

### Tricks

Take, when needed, as three consecutive terms of an A.P. $(\alpha – \beta), \alpha , (\alpha + \beta)$
And, when four is for asking: $(\alpha – 3\beta), (\alpha – \beta), (\alpha + \beta), (\alpha + 3\beta)$

## Geometric Progression

G.P or G.S is a sequence in which the ratio of any term and the preceding term is always a constant. Symbolically, $\frac{t_n}{t_{n – 1}} = k, \text{ for }n \geq 2$

### Properties:

• G.S is completely known if one term and the common ratio $$r$$ are known.
• Multiply/divide all terms with the same quantity, we get a new G.S.
• The product of the corresponding terms of two G.S results in a new G.S.
• Reciprocals of all the terms make a new G.S.

Symbolically, Let given G.S be: $a + ar + ar^2 + ar^3 + ⋯ +ar^{n – 1}$ Then, these are also in G.S: $ab + abr + abr^2 + ⋯ +abr^{n – 1},\text{ with }t_1 = ab\text{ and }\text{c.r. } = r$ $\frac{a}{b} + \frac{ar}{b} + \frac{ar^2}{b} + ⋯ +\frac{ar^{n – 1}}{b},\text{ with }t_1 = \frac{a}{b}\text{ and c.r. } = r$ $\frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + ⋯, \text{ with }t_1 = \frac{1}{a}\text{ and c.r. } = \frac{1}{r}$

### Formulae

$t_n = ar^{n – 1}$ $S_n = \frac{a(1 – r^n)}{1 – r}$ $S_{\propto } = \frac{a}{1 – r}, \text{ where }|r| < 1$ $\text{ G.M } = \sqrt{ab}$

Three consecutive terms: $\frac{a}{r}, a, ar$

Four consecutive terms: $\frac{a}{r^3}, \frac{a}{r}, ar, ar^3$

## Harmonic Progression

Harmonic series is formed by the reciprocals of an arithmetic series. General form: $\frac{1}{a} + \frac{1}{a + d} + \frac{1}{a + 2d} + ⋯$ To find $$t_n$$ of H.P, we follow the A.P route. No general formula to sum up an H.P.
Harmonic Mean and relations between respective means: A, G and H are in G.P. $H.M. = \frac{2ab}{a + b}, \text{ where } a \neq -b$ $A \times H = G^2$ $A > G > H$

Wikipedia