In this session explained about Basic concepts of Sequence and series, introduction and formulas for *arithmetic progression* like Common Difference, Finite and infinite arithmetic Progressions, **n ^{th}** term , sum to first

**n**term and Properties of Arithmetic Progression.

^{th}## Arithmetic Progression Definition, Properties, Formulas | Allmathtricks

Sequence following specific patterns are called progressions.

Basic concepts of sequence and series

Progressions are of three types. They are

1. Arithmetic Progression

2. Geometric Progression

3. Harmonic Progression

### Arithmetic Progression

Arithmetic Progression can be defined as, a sequence of numbers is obtained by adding a fixed number “**d”** to the preceding term except the first term.

i.e A list of numbers X_{1}, X_{2}, X_{3}, X_{4} . . . . . . . is an A.P, If the differences X_{2}-X_{1}, X_{3}-X_{2}, X_{4}-X_{3}, . . . . given the same value

Each of the number in the list of an **arithmetic progression** is called a term of that A.P.

The general form of an A.P is **a, a+d, a+2d, a+3d . . . . . .** ( Here “**d**” is the common difference and “**a**” is the first term )

#### Common Difference

The difference between two succeeding terms of an __Arithmetic Progression __ is called *common difference*. The difference value can be positive,negative or zero.

**Examples:**

1) Rainfall of the last week ( in mm) 12, 10, 08, 06, 04, 02, 0

2) Temperature record for the last week ( in Deg.C) 30, 31, 32, 33, 34, 35, 36

3) 2, 4, 6, 8, 10, . . . . . . . . . . . .

4) 0, -5, -10, -15, -20, . . . . . . . . . . .

**Finite and infinite arithmetic Progressions**

In the above examples no. 1 & 2 having finite number of terms. Such arithmetic Progressions is called **finite A.P.** In the same way above examples no. 3 & 4 having infinite number of terms. So they are called** infinite arithmetic Progressions**

**Formula for finding the n ^{th} term or general term of AP formula
**

Let r_{1}, r_{2}, r_{3}, r_{4}, . . . . . be an AP, Here first term ‘r_{1}‘ is “r” and common difference is “d”

Then,

First term r_{1} = r = r + (1 -1 ) d

Second term r_{2} = r + d = r + ( 2-1)d

Third term r_{3} = r_{2} + d = r + ( 2-1)d +d = r + (3 -1) d

Fourth term r_{4} = r_{3} + d = r + (3 -1) d +d = r + (4 -1) d

Firth term r_{5 }= r_{4} + d = r + (4 -1) d +d = r + (5 -1) d

Similarly use same pattern for **n ^{th}** term

**r _{n} = r + (n -1) d**

**r _{n} **is also called

**general term of the A.P**

Using the above formula we can find different terms of the A.P

General notation of **n ^{th}** terms of an

*Arithmetic progression*

**Tn = a + ( n-1) d**

Here Tn = **n ^{th}** of an A.P ;

**a**= first term of the A.P ;

**d**= common difference

If number of terms in an A.P is “** r”** and last term is “**L**” then,

**L = a + ( r – 1)d**

#### Formula for finding the sum to first n^{th} term an A.P

Let **a, a+d, a+2d, a+3d, . . . . . . . . . ** be an AP, Here first term **“a**” and common difference is **“d**‘

The **n ^{th}** term of this A.P is

**a**= a + ( n-1) d

_{n}Let Sum of “n” terms of an A.P is ” **Sn**“

**Sn** = a + (a+d) + (a+2d) + ( a+3d) + . . . . . . . . . + [ a + ( n-1) d ]

Now write Sn from last term of A.P

**Sn** = [ a + ( n-1) d ] + [ a + ( n-2) d ] + [ a + ( n-3) d ] + . . . . . . . . . . . + a

Adding Sn +Sn

⇒ 2Sn = [ 2a + ( n-1) d ] + [ 2a + ( n-1) d ] + [ 2a + ( n-1) d ] +. . . . . . . . . . .+ [ 2a + ( n-1) d ] ( n times)

⇒ 2Sn = n [ 2a + ( n-1) d ]

⇒

⇒ Sn = (n/2) [ a + a + ( n-1) d ] = (n/2) ( first term + **n ^{th}** term)

⇒ **Sn = (n/2) ( a + a _{n})**

### Properties of Arithmetic Progression

**1. ****Reversed Sequence**

If the Arithmetic Progression a, a+d, a+2d, a+3d . . . . . . m is reversed to m, m-d , m-2d, m-3d, . . . . a, then the common difference changes to negative of original common difference, and the reversed sequence is also A.P.

i.e a_{1}, a_{2}, a_{3},, . . . . . a_{n-1}, a_{n }are in A.P then a_{n}, a_{n-1}, . . . . . a_{3}, a_{2}, a_{1}, are in A.P

**Example:**

1, 3, 5, 7, 9 is an A.P with common difference ” 2″

Now 9, 7, 5, 3, 1 is also an A.P with common difference “-2 “

**2.** If the Arithmetic Progression a, a+d, a+2d, a+3d . . . . . . , then

If d >0, then a_{n} > a_{n-1 } ( Here a_{n} is a n^{th} term of above A.P)

If d < 0, then a_{n} < a_{n-1 } ( Here a_{n} is a n^{th} term of above A.P)

**3.** If the sum of the first two terms is “a” and the sum of last two terms is “b” , number of terms is “n”

then the sum of the A.P is

**4. ** The **n ^{th}** term of Arithmetic Progression is the difference of the sum to first “n” terms and sum of first (n-1) terms of it.

i.e a_{n} = Sn – Sn-1

**5.** If r_{1}, r_{2}, r_{3}, r_{4}, . . . . . r_{n} be an finite A.P, then the sum of the terms equidistant from the beginning and the end is always same and is equal to the sum of the first and last term.

i.e r_{1} + r_{n }= r_{2} + r_{n-1 }= r_{3} + r_{n-2 }= r_{4} + r_{n-3 }= r_{5} + r_{n-4
}

**6.** If r_{1}, r_{2}, r_{3}, r_{4}, . . . . . r_{n} be an finite A.P, then

i) r_{n}, r_{n-1}, r_{n-3, } . . . . . r_{2, }r_{1 }are in A.P

ii) r_{1} ± k, r_{2} ± k, r_{3 }± k, r_{4}± k, . . . . . r_{n }± k are in A.P ( Where k ∈ R )

iii) kr_{1}, kr_{2}, kr_{3}, kr_{4}, . . . . . kr_{n} are in A.P ( Where k ∈ R – {0} )

**7. ** If r_{1}, r_{2}, r_{3}, r_{4}, . . . . . r_{n} & a_{1}, a_{2}, a_{3}, a_{4}, . . . . . a_{n} are two finite A.P, then

i) a_{1}+r_{1}, a_{2}+r_{2}, a_{3}+r_{3}, . . . . . a_{n}+r_{n}_{ }is in A.P

ii) a_{1}r_{1}, a_{2}r_{2}, a_{3}r_{3}, . . . . . a_{n}r_{n}_{ }is not in A.P

iii) a_{1/}r_{1}, a_{2/}r_{2}, a_{3/}r_{3}, . . . . . a_{n/}r_{n}_{ }is not in A.P

**Related Articles**

Arithmetic Progression Formulas

Arithmetic Progression Examples

Arithmetic Mean formula with Examples

Geometric Progression formulas

Geometric Progression Examples

Relationship Between AM, GM and HM

Factorizing Algebraic Expressions