Sequences and Series

Define sequences and series and understand the classification of sequences (finite and infinite). nth terms of sequences and series

Sequences

An arrangement of numbers in a definite order following some rule is known as a sequence. We also define a sequence as a function whose domain is the set of natural numbers or some subset of the type {1, 2,…, k}.

For example: 6, 12, 18, 24…; n, n + 1, n + 2, n + 3, n + 4, n + 5; etc.

In general, a sequence is denoted by {an} or < an> which represents the sequence a1, a2, a3,… an.

The numbers a1, a2, a3 … and an occurring in a sequence are called its terms, where the subscript denotes the position of the term. The nth term or the general term of a sequence is denoted by an.

There are two types of sequences: finite and infinite.

A sequence containing finite number of terms is called a finite sequence. For example: 5, 10, 15, 20, 25, 30, 35 is a finite sequence. A sequence containing infinite number of terms is called an infinite sequence. For example: sequence of prime numbers, sequence of natural numbers etc. are infinite sequences.

Note: Sometimes, a sequence is denoted by {Tn} or < Tn> which represents the sequence T1, T2, T3,… Tn.

Sum of first n terms of a sequence:

Let {an} be the sequence such as {an} = a1, a2, a3,… an. Also, let Sn be the sum of its first n terms.

Then we have

Sn = a1 + a2 + a3 +…+ an

It can be be observed that:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

.       .      .       .

.       .      .       .

.       .      .       .

Sn = a1 + a2 + a3 +…+ an

From the above equations, we obtain

S1 = a1

S2 – S1= a2

S3 – S2= a3

.       .      .  

.       .      .  

.       .      .   

Sn – Sn – 1= an

⇒ an = Sn – Sn – 1

So, if Sn is known then any term of the sequence can be obtained.

Fibonacci sequence:

If a sequence is generated by a recurrence relation, each number being the sum of the previous two numbers, then it is called a Fibonacci sequence. For example: a1 = a2 = 1, a3 = a1 + a2, an = an−2 + an−1, n > 2.

Let's now try and solve the following puzzle to check whether we have understood the concept of Fibonacci sequence.

 

Series

If a1, a2, a3… an is a given sequence, then the expression a1 + a2 + a3 + …+ an is called the series associated with the sequence.

For example: The series associated with the sequence 18, 36, 54, 72, 90 … is 18 + 36 + 54 + 72 + 90 + … The series associated with the sequence 2, 4, 6, 8, 10, 12 is 2 + 4 + 6 + 8 + 10 + 12 In a compact form, the series associated with the sequence a1, a2, a3, …, an can be written in sigma notation as, where sigma ( ∑ ) denotes the sum. Note that series a1 + a2 + a3 + …+ an does not refer to the actual sum of the numbers a1, a2, … , an. Infact, it just refers to the indicated sum, and it shows that a1 is the first term, a2 is the second term, …, an is the nth term of the series. The series is finite or infinite depending on the given sequence.

For example: The series associated with the sequence of numbers that are multiples of 4 is an infinite series, whereas the series associated with the sequence of numbers that are odd and less than 100 is a finite series.

 

Solved Examples

Example 1:

The nth term of a sequence is given by an =. Find the ratio of the 6th term of the sequence to its 4th term.

Solution:

The nth term of a sequence is given by an =.

Therefore,

Thus, the ratio of the 6th term of the sequence to its 4th term is given by

Hence, the required ratio is 129:17.

Example 2:

Write the first five terms of the sequence whose nth term is given by.

Solution:

It is given that the nth term of the sequence is given by.

Hence, an =.

On putting n = 1, 2, 3, 4, 5 successively in an, we obtain

a1 =

a2 =

a3 =

a4 =

a5 =

Thus, the required first five terms of the sequence are.

Example 3:

Write the first six terms of the series associated with the following sequence:

a1 = 5 a2 = −3, an = an−1 + an−2, n > 2.

Solution:

The sequence is given by

a1 = 5 a2 = −3

an = an−1 + an−2, n > 2

Substituting n = 3, 4, 5, 6, we obtain

a3 = a2 + a1 = −3 + 5 = 2

a4 = a3 + a2 = 2 + (−3) = 2 − 3 = −1

a5 = a4 + a3 = (−1) + 2 = 1

a6 = a5 + a4 = 1 + (−1) = 0

Thus, the first six terms of the given sequence are 5, −3, 2, −1, 1, 0.

Thus, the first six terms of the series associated with the given sequence is 5 + (−3) + 2 + (−1) + 1 + 0.

 

Example 4:

The nth term of a sequence is given by. Then show that

 

Answer:

Put n = 5, 10 and15 successively in Tn to obtain T5, T10 and T15 respectively.

 

 

Example 5:

The nth term of a sequence is given by. If , then prove that n =7.

 

Answer: Tn+2Tn=2n+2-52n-5=139⇒2n-12n-5=139⇒26n-65 = 18n-9⇒8n = 56⇒n = 7Therefore, n=7

Sequences

An arrangement of numbers in a definite order following some rule is known as a sequence. We also define a sequence as a function whose domain is the set of natural numbers or some subset of the type {1, 2,…, k}.

For example: 6, 12, 18, 24…; n, n + 1, n + 2, n + 3, n + 4, n + 5; etc.

In general, a sequence is denoted by {an} or < an> which represents the sequence a1, a2, a3,… an.

The numbers a1, a2, a3 … and an occurring in a sequence are called its terms, where the subscript denotes the position of the term. The nth term or the general term of a sequence is denoted by an.

There are two types of sequences: finite and infinite.

A sequence containing finite number of terms is called a finite sequence. For example: 5, 10, 15, 20, 25, 30, 35 is a finite sequence. A sequence containing infinite number of terms is called an infinite sequence. For example: sequence of prime numbers, sequence of natural numbers etc. are infinite sequences.

Note: Sometimes, a sequence is denoted by {Tn} or < Tn> which represents the sequence T1, T2, T3,… Tn.

Sum of first n terms of a sequence:

Let {an} be the sequence such as {an} = a1, a2, a3,… an. Also, let Sn be the sum of its first n terms.

Then we have

Sn = a1 + a2 + a3 +…+ an

It can be be observed that:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

.       .      .       .

.       .      .       .

.       .      .       .

Sn = a1 + a2 + a3 +…+ an

From the above equations, we obtain

S1 = a1

S2 – S1= a2

S3 – S2= a3

.       .      .  

.       .      .  

.       .      .   

Sn – Sn – 1= an

⇒ an = Sn – Sn – 1

So, if Sn is known then any term of the sequence can be obtained.

Fibonacci sequence:

If a sequence is generated by a recurrence relation, each number being the sum of the previous two numbers, then it is called a Fibonacci sequence. For example: a1 = a2 = 1, a3 = a1 + a2, an = …