# 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'.

• nth number is denoted as 'Tn'.

• Sum of n number is denoted as 'Sn'.

### 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.

```Tn = 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.

```Sn = (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:

```Sn = (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'.

• nth number is denoted as 'Tn'.

• Sum of n number is denoted as 'Sn'.

### 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.

```Tn = 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.

```Sn = a(1 - rn)/(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:

```Sn = a(rn - 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)
```

```12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6
```

```13 + 23 + 33 + ... + n3 = [(1/2)n(n+1)]2
```