Difference between Sequence and Series: Overview, Questions, Preparation

To understand the pattern of the list of numbers, we often use the sequences and series of the terms. Though they may sound similar, many vital differences make these listings unique. 

Sequence 

When a list of numbers is arranged or sorted in a particular order, it is called a sequence. The numbers in a sequence are called terms, and each term is separated from the other with a common difference. The end of the sequence is usually referred to by three connected dots that tells that the sequence is continuous. 

For instance, 3, 6, 9, 12, 15,… is a sequence where each successive number is greater than the previous number by 3. There are 4 types of sequences in mathematics viz: Fibonacci Sequence, Harmonic Sequence, Arithmetic Sequence, and Geometric Sequence. 

Series is a list of numbers that is presented as a sum of the numbers that are in a sequence. Therefore, the addition symbol ‘+’ is placed between every two numbers in a series. A series is said to be finite if it represents a finite sequence, and it is said to be infinite if it represents an infinite sequence. 

For instance, 2+4+6+8… is an infinite series, whereas 3+5+7+9 is a finite series. There are 5 different types of series viz. Exponent Series, Harmonic Series, Geometric Series, Alternating Series, and Power Series. 

Difference between Sequence and Series 

The differences between Sequence and Series discussed below:

Pattern: In a sequence, the elements adhere to a particular pattern that defines it. However, since series is an addition of the elements of a sequence, it may or may not follow a particular pattern. 

Order: The basic difference between sequence and series is that the order of elements is important for sequence. However, the order of the elements does not matter when it comes to series.

For example, 3, 4, 5 and 4, 5, 3 are different sequences, but when it comes to series, 3+4+5 is equal to 4+5+3. 

Finiteness/Infiniteness: A sequence may be finite or infinite depending upon how it is defined. However, a series is finite if the sequence is a list of finite numbers, and if the sequence is infinite, then the series is also said to be infinite. 

The difference between series and sequence is clearly explained in the chapter ‘Sequence and Series’ of Class XI. It carries a weightage of 8 to 9 marks in the exam. 

1. Sequence S = 1, 3, 5, 7…. Find its 21^st term.

Solution. The common difference, i.e., d, is 2.

The n^th term can be found by the formula: Tn = a + (n-1)2 = 1 + (21-1)2 = 1 + 40 = 41

2. The arithmetic progression and geometric progression of two numbers are 8 and 10. Find the numbers.

Solution.

A.M = 8    

G.M = 10

Let the numbers be a and b

Arithmetic progression is given by the formula = (a + b)/2

Geometric progression is given by the formula = √ab 

As per the given values, (a + b)/2 = 8

a + b = 16 --(1)

√ab = 10

ab = 100

Now, we have the formula, (a-b)2 = (a+b)2 - 4ab = 256 - 400 = - 144

a - b = -12 ---(2)

From 1 and 2 we get a = 2 and b = 14

Q: Define arithmetic sequence.

Q: The concept of sequence and series is relevant to which fields?

Q: Why do the elements of a series not follow a particular order?

Q: Who invented the concept of sequence and series?

Q: How can the nth term of a sequence be described?