Math-for-all-Grades

Click on any of the following links to learn Sequences and Series in detail:

** How do you denote the terms of a Sequence?**

** How do you denote terms in an Arithmetic Sequence?**

** What is General term in an Arithmetic Sequence? **

** How to find sum of n terms of an Arithmetic Sequence?**

** How do you denote terms in a Geometric Sequence? **

**What is General Term in a Geometric Sequence? **

How to find sum of n terms of a Geometric Sequence?

A sequence is a list of numbers written in order.

(Sequence is also called Progression)

The following is a sequence of odd numbers:

1, 3, 5, 7, 9...

The numbers of a sequence are called the Terms of the sequence

In the above sequence of **odd numbers,**

The First term is 1, the Third term is 5 and so on.

**Terms** of a sequence are denoted using letters and subscripts.

a_{1} , a_{2} , a_{3}, a_{4}, ..........., a_{n} are the symbols used for denoting terms of a sequence.

a_{1} is the first term, a_{2} is the second term and an is the last term of the sequence.

**Series ** is sum of the terms of sequence.

The two main types of Sequences are:

Arithmetic Sequence and Geometric Sequence

Numbers are said to be in Arithmetic Sequence if there is a common difference between any two consecutive terms.

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

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 1st term,

a + d is 2nd term,

a + 2d is 3rd 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.

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**.

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 nth term and it is denoted as **an**

To find this **an** 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 50th 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

an = a + (n – 1)d.

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

Applying the above formula, the 50th term is

3 + (50 – 1) × 4 = 3 + 196 = 199

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

**Solution: **

Represent 25th term as a25.

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

we can write, a25 = 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 15th term is

a15 = 60 + (15 – 1) × 5 = 130

Series is defined as sum of the terms of a sequence.

Normally, sum of n terms of a sequence is denoted as **S _{n} **

If **a _{1} , a_{2} , a_{3}, a_{4}, ..........., a_{n}** are the n terms of a sequence, then

**S _{n} = a_{1} + a_{2} + a_{3} + a_{4} + ........+ a_{n} **

**How to find the sum of n terms of an Arithmetic Sequence: **

Let the n terms of an Arithmetic Sequence be

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

If Sn is the sum of the above n terms which are in Arithmetic Sequence, then

Sn = n/2 [2a + (n – 1 ) d]

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

Since l = a + (n – 1)d, therefore

Sn = 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