Arithmetic Sequence:

An arithmetic sequence is defined as a group of numbers in which any two consecutive or successive terms have a common difference.

**Example: **

In the sequence of following numbers:

3, 7, 11, 15, 19, ....

the common difference between any two **consecutive numbers** is 4.

Therefore, the numbers 3, 7, 11, 15, 19 ...are said to be in Arithmetic Sequence

**Denotation of terms in an Arithmetic Sequence**

In an Arithmetic Sequence, the following denotations are generally followed:

**a** denotes the first term,

**d**, the common difference and

**n**, the position of any term or a particular term.

Then the **n **terms of an **Arithmetic Sequence **are written as

**a, a +d, a + 2d, a + 3d, a + 4d, ............., a + (n – 1)×d**

In the above sequence of **n** terms

a is 1^{st} term,

a + d is 2^{nd} term,

a + 2d is 3^{rd} term, and

a + (n – 1) ×d is the **last **term

**a + (n – 1) × d** is also called the last term or the **n ^{th} ** term or still the

We will discuss below the formulae for finding any particular term and sum of any number of terms in an arithmetic sequence.

**What is n ^{th} term of a sequence?**

The nth term of a sequence denotes the position of a term or a particular term in the sequence.

Any term that is to be found is referred to as **n ^{th} term or general term**.

How to find the n^{th} term/general term of an Arithmetic Sequence?

Let the n terms of an Arithmetic Sequence with a **common difference ‘d’** be

a, a + d, a +2d, ........ a + (n – 1) d

Any term in an Arithmetic Sequence is called general term or n^{th} term and it is denoted as **a _{n}**

To find this **a _{n}** the following formula is applied:

**a _{n} = a + (n – 1) d**, where

a is the first term,

d is the common difference and

n is the position of the term.

**Example**

1. The first three terms of an Arithmetic Sequence are 3, 7, 11 ...What is the 50^{th} term?

**Solution: **

In the A.S. 3, 7, 11, the common difference is 4.

To find the 50th term, apply the above formula for nth term

a_{n} = a + (n – 1)×d.

In this the first term a = 3, common difference d = 4 and n = 50.

Applying the above formula, the 50^{th} term is

3 + (50 – 1) × 4 = 3 + 196 = 199

2. In an arithmetic sequence with a common difference of 5, the 25^{th} term is 180. What is the 15^{th} term of this sequence?

**Solution: **

Represent 25^{th} term as a_{25}.

Next, from the n^{n} term formula for an A.S.: a_{n} = a + (n – 1) ×d,

we can write, a_{25} = 180, n = 25, d = 5.

What needs to be found is the first term, a.

Applying the nth term formula, we get

180 = a + (25 – 1) × 5,

180 = a + 120, so, a = 60.

Again, from the nth term formula, the 15^{th} term is

a_{15} = 60 + (15 – 1) × 5 = 130

Let the n terms of an Arithmetic Sequence be

a, a + d, a + 2d, ............, a + (n – 1)× d

If S_{n} is the sum of the above n terms which are in Arithmetic Sequence, then

S_{n} = n/2 [2a + (n – 1 ) d]

Now, let ‘l’ denote the last term, then

Since l = a + (n – 1)d, therefore

S_{n} = n/2[a + l]

Example:

**1. The terms of an Arithmetic Sequence are 2, 5, 8, 11, ..........**

**What is the sum of the first 51 terms of this sequence? **

**Solution:**

Now, first term, a = 2,

Common difference, d = 3,

and the last term is

l = a + (n – 1) × d is

2 + (51 – 1) × 3 = 2 + 150 = 152

Therefore, S = 51/2 [2 + 152] = 3927

** Arithmetic progression. **

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