# Formulas for Arithmetic Progression

## Formulas To Solve Arithmetic Progression Questions in Aptitude

An Arithmetic progression or Arithmetic  sequence is a Sequence of numbers such that the difference of any two successive members is a constant.In Many Examinations Questions Comes from Arithmetic progression Topic that’s why we need to Remember Formulas for Arithmetic Progression.

In this Page All Formulas for Arithmetic Progression is given.That is Very important to solve any problems of Arithmetic progression .

### Basic Concept on Arithmetic Progression

First term is denoted by a

Common difference is denoted by d

nth term is denoted by $a_{n}$ or $t_{n}$

Sum of First n terms is denoted by $S_{n}$

Example : 4,8,12,16……..

### Finite or Infinite Arithmetic Progressions

• Finite Arithmetic Progression

When there are limited number of terms in the sequence then it is known as Finite Arithmetic Progression.

For example: 10, 20, 30, 40, 50

• Infinite Arithmetic Progression

When there are unlimited number of terms in the sequence then it is known as Infinite Arithmetic Progression.

For example: 3, 5, 7, 9, 11, 13, ..…….

### Formula of Arithmetic Progression

• nth term of an AP

Formula to find the nth term of an AP is

Tn = a + (n – 1) d

where tn = nth term,

a= first term ,

d= common difference,

n = number of terms in the sequence.

### Number of terms in an AP

• Formula to find the numbers of term of an AP is

$n = \left [ \frac{(l-a)}{d} \right ] + 1$

where

n = number of terms,

a = first term,

l = last term,

d= common difference.

### Sum of first n terms in an AP

• Formula to find the sum of first n terms of an AP is

$S_{n} = \frac{n}{2} [2a + (n-1)d]$

OR

$S_{n} = \frac{n}{2} (a+l)$

where,

a = first term,
d= common difference,
tn = nth term = a + (n-1)d

### Arithmetic Mean

If a, b, c are in AP, then the Arithmetic mean of a and c  is b  i.e.

$b = \frac{1}{2} (a + c)$

### Some other important formulas of Arithmetic Progression

• Sum of first n natural numbers

We derive the formula to find the sum of first n natural numbers

$S = \frac{n (n+1)}{2}$

where

S = Sum of first n natural numbers

n = number of First n natural numbers

### Sum of squares of first n natural numbers

• Formula to find the sum of squares of first n natural numbers is

$S= \frac{{ n (n+1) (2n+1) }} {6}$

where

S = Sum of Squares of first n natural numbers

n = number of First n natural numbers.

### Sum of first n odd numbers

• Formula to Find the Sum of First n odd numbers

S = n2

where

S = Sum of first n odd  numbers

n = number of First n odd numbers.

### Sum of first n even numbers

• Formula to find the Sum of First n Even numbers is

S = n(n+1)

where

S = Sum of first n Even numbers

n = number of First n Even numbers.

### Properties of Arithmetic Progression

• If a fixed number is added or subtracted from each term of an AP, then the resulting sequence is also an AP and it has the same common difference as that of the original AP.
• If each term in an AP is divided or multiply with a constant non-zero number, then the resulting sequence is also in an AP.
• If nth term is in linear expression then the sequence is in AP.
• If a1, a2, a3, …, an and b1, b2, b3, …, bn, are in AP. then a1+b1, a2+b2, a3+b3, ……, an+band a1–b1, a2–b2, a3–b3, ……, an–bn will also be in AP.
• If nth term of a series is Tn = An + B, then the series is in AP
• Three terms of the A.P whose sum or product is given should be assumed as a-d, a, a+d.
• Four terms of the A.P. whose sum or product is given should be assumed as a-3d, a-d, a+d, a+3d.