## Aptitude – Basic Arithmetic

Tutorials to solve Basic Arithmetic Questions in Aptitude

Arithmetic is the fundamental of mathematics that includes the operations of numbers. These operations are addition, subtraction, multiplication and division. Arithmetic is one of the important branches of mathematics, that lays the foundation of the subject ‘Maths’, for students.

## Basic Arithmetic

The basic operations under arithmetic are addition and subtraction, division and multiplication, although the subject involves many other modified operations.

### What is Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers, where the difference between one term and the next is a constant. For example, 1, 4, 7, 10, 13, 16, 19, 22, 25, … is an arithmetic sequence with common difference equal to 3. It is also termed arithmetic progression and commonly represented as:

`a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d`

Where, a = first term, d = common difference between the terms & n = number of terms

## Arithmetic 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 Formulas

1]. 1 + 2 + 3 + …. + n = (1/2)n(n+1)
2]. 12 + 22 + 32 + …. + n2 = n(n+1)(2n+1)/6
3]. 13 + 23 + 33 + …. + n3 = [(1/2)n(n+1)]2