# Aptitude - Basic Arithmetic

## Sequence

A sequence represents numbers formed in succession and arranged in a fixed order defined by a certain rule.

## Airthmetic Progression ( A.P.)

It is a type of sequence where each number/term(except first term) differs from its preceding number by a constant. This constant is termed as common difference.

### A.P. Terminologies

First number is denoted as 'a'.

Common difference is denoted as 'd'.

n

^{th}number is denoted as 'T_{n}'.Sum of n number is denoted as 'S

_{n}'.

### A.P. Examples

1, 3, 5, 7, ... is an A.P. where a = 1 and d = 3 - 1 = 2.

7, 5, 3, 1, - 1 ... is an A.P. where a = 7 and d = 5 - 7 = -2.

### General term of A.P.

T_{n}= a + (n - 1)d

Where **a** is first term, **n** is count of terms and **d** is the difference between two terms.

### Sum of n terms of A.P.

S_{n}= (n/2)[2a + (n - 1)d

Where **a** is first term, **n** is count of terms and **d** is the difference between two terms. There is another variation of the same formula:

S_{n}= (n/2)(a + l)

Where **a** is first term, **n** is count of terms, **l** is the last term.

## Geometrical Progression, G.P.

It is a type of sequence where each number/term(except first term) bears a constant ratio from its preceding number. This constant is termed as common ratio.

### G.P. Terminogies

First number is denoted as 'a'.

Common ratio is denoted as 'r'.

n

^{th}number is denoted as 'T_{n}'.Sum of n number is denoted as 'S

_{n}'.

### G.P. Examples

3, 9, 27, 81, ... is a G.P. where a = 3 and r = 9 / 3 = 3.

81, 27, 9, 3, 1 ... is a G.P. where a = 81 and r = 27 / 81 = (1/3).

### General term of G.P.

T_{n}= ar^{(n-1)}

Where **a** is first term, **n** is count of terms, **r** is the common ratio

### Sum of n terms of G.P.

S_{n}= a(1 - r^{n})/(1 - r)

Where **a** is first term, **n** is count of terms, **r** is the common ratio and r < 1. There is another variation of the same formula:

S_{n}= a(r^{n}- 1)/(r - 1)

Where **a** is first term, **n** is count of terms, **r** is the common ratio and r > 1.

## Arithmetic Mean

Airthmetic mean of two numbers a and b is:

Arithmetic Mean = (1/2)(a + b)

## Geometric Mean

Geometric mean of two numbers a and b is

Geometric Mean = √ab

## General Formulaes

1 + 2 + 3 + ... + n = (1/2)n(n+1)

1^{2}+ 2^{2}+ 3^{2}+ ... + n^{2}= n(n+1)(2n+1)/6

1^{3}+ 2^{3}+ 3^{3}+ ... + n^{3}= [(1/2)n(n+1)]^{2}