Sequences and Series

 

  • What is a Sequence?
  • How do you denote the terms of a Sequence?
  • What is Arithmetic 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?
  • What is Geometric 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?
  • What is a 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.


 

Denotation of Terms in a Sequence:

Terms of a sequence are denoted using letters and subscripts. a1 , a2 , a3, a4, .………., an are the symbols used for denoting terms of a sequence. a1 is the first term, a2 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.

The Arithmetic 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

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 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 nth term or still the general term of the above arithmetic sequence.

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

What is nth 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 nth term or general term.

How to find the nth 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 nth term and it is denoted as an

To find this an the following formula is applied:

  • an = 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:

Series is defined as sum of the terms of a sequence. Normally, sum of n terms of a sequence is denoted as Sn

If a1 , a2 , a3, a4, ……….., an are the n terms of a sequence, then Sn = a1 + a2 + a3 + a4 + ……..+ an

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