Introduction to Sequences and Series

Imagine yourself at a pizza hut.  You have placed an order and your order number is 282. So currently they are serving the order number 275. So how many orders do you think will be served before your number? Yes, six orders more because you are in a sequence. To understand this better, let us learn about the sequences and series. Let us do it right now.

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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 a sequence in which the number of terms is countable, they are called finite sequences. A finite sequence has a finite number of terms. So as discussed earlier here 1 is the term, 3 is the term so is 5, 7, …..

Browse more Topics under Sequences And Series

• Arithmetic Progression
• Geometric Progression
• Means
• Special Series

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 a_n here ‘ n ‘ is the n^th 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 a₃ will be given as a₁  + a₂

Therefore, 1 + 1 = 2. Similarly,
a_{4 =} a₂ + a₃
∴ 1 + 2 = 3
a_5 = 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 for You

Question: Identify the sequence of the following function n (n+3)

1. 4, 10, 26, …
2. 4, 12, 18, …
3. 2, 10, 16, …
4. 4, 10, 27, …

Solution: Correct option is A. The given function is n(n+3),

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

1. term
2. series
3. constant
4. 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.