Introduction to Sequences and Series | Mathematics (Maths) for JEE Main & Advanced PDF Download

Sequences

A sequence is a list of numbers in a special order. It is a string of numbers following a particular pattern, and all the elements of a sequence are called its terms. Let us consider a sequence,

[1, 3, 5, 7, 9, 11……]

We can say this is sequence because we know that they are the collection of odd natural numbers. Here the number of terms in the sequence will be infinite. Such a sequence which contains the infinite number of terms is known as an infinite sequence. But what if we put end to this.

[1, 3, 5, 7, 9, 11…..131]

If 131 is the last term of this sequence, we can say that the number of terms in this sequence is countable. So in such sequence in which the number of terms is countable, such sequences are called finite sequences. A finite sequence has the finite number of terms. So as discussed earlier here 1 is the term, 3 is the term so is 5, 7…..

Fibonacci Sequence
The special thing about the Fibonacci sequence is that the first two terms are fixed. When we talk about the terms, there is a general representation of these terms in sequences and series. A term is usually denoted as an here ‘ n ‘ is the nth term of a sequence. For the Fibonacci sequence, the first two terms are fixed.

The first term is as a₁= 1 and a₂= 1. Now from the third term onwards, every term of this Fibonacci sequence will become the sum of the previous two terms. So a3will be given as a₁  + a₂

Therefore, 1 + 1 = 2. Similarly,
a₄ = a₂ + a₃
∴ 1 + 2 = 3
a₅ = a₃+ a₄
∴ 2 + 3 = 5

Therefore if we want to write the Fibonacci sequence, we will write it as, [1  1  2  3  5…]. So, in general, we can say,

a_n = a_n-1 + a_n-2

where the value of n ≥ 3.

Types of Sequences

• Arithmetic sequence: In an arithmetic (linear) sequence the difference between any two consecutive terms is constant.

• Quadratic Sequence: A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant.

• Geometric Sequence: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Series
Series is the sum of sequences. The series is finite or infinite according to the given sequence is finite or infinite. Series are represented as sigma, which indicates that the summation is involved. For example, a series S can be,
S = Sum (1, 3, 5, 7, 9, 11, …)

Solved Examples
Question: Identify the sequence of the following function n (n+1)
4, 10, 18, 28…
4, 12, 18, 28…
2, 10, 18, 28…
4, 10, 18, 28…
Solution: Correct option is A. The given function is n(n+1),
When n = 1, 1(1+3) = 4
n = 2, 2(2+3) = 10
n=3, 3(3+3) = 27
So, 4, 10, 27…is the function for the sequence n(n+3).

Question: Adding first 100 terms in a sequence is
A. term
B. series
C. constant
D. sequence
Solution: Correct option is B. Adding first 100 terms in a sequence is series. Also adding the number of some set is a series.