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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

S2S1= a2

S3S2= a3

.       .      .

.       .      .

.       .      .

SnSn – 1= an

⇒ an = SnSn – 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
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• 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.
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• 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.
• !--
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• 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…

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