Sequence and Series – Difference, Definitions

Sequence and series are crucial topics in mathematics. A sequence is a  list of elements in which repetitions of any sort are allowed, whereas the sum of all elements is known as a series. For example, 2, 4, 6, 8, 10 is a sequence with five elements and the corresponding series will be 2+4+6+8+10, where the sum of the series will be 30. Now let us understand them.  An AP(arithmetic progression) is one example of sequence and series.

Sequence: 

It is defined as the collection of numbers which are arranged in a specific pattern where each number in the sequence is known as a term. 

For example- 5, 10, 15, 20, 25, … is a sequence where the ellipsis sign at the end of the sequence represents that the list continues further till infinity. In this sequence 5 is the first term, 10 is the second term,15 is the third term and so on. Each component in the sequence has a common difference, and the order continues with the common difference. In the example given above, the common difference is 5. The types of sequences present are:

• Arithmetic Sequence
• Geometric Sequence
• harmonic Sequence
• Fibonacci Sequence

Types of Sequences

1. Arithmetic Sequence 

It is a sequence in which every term is found by adding or subtracting a definite number(common difference) from the preceding number.

x₁,  x₁+ m , x₁+ 2m, x₁+ 3m,….. x₁+ (n-1)m and so on, where m is the common difference.

For example- 1, 4, 7, 10,13,16…….

Here, we observe that the common difference is 3. 

2. Geometric Sequence

A sequence in which each of the terms in a series is obtained by either multiplying or dividing a constant number with the former one is said to be a geometric sequence. The common ratio of a GP is obtained by taking the ratio between any one term in the sequence and dividing it by the previous term.

For example- 2, 4, 8, 16 ,32 …

Here, the common ratio or factor is 2.

General form- a, am, am², am³, am⁴,…, amⁿ 

where, a is the first term and m is the common ratio between the terms.

3. Harmonic Sequences

A harmonic sequence is a sequence where the reciprocals of all the terms of the sequence form an arithmetic sequence.

4. Fibonacci Number Sequence

Fibonacci numbers form a sequence of numbers in which each of the terms is obtained by adding two preceding terms. The sequence starts with 0 and 1.

For example- 

F₁ = 0, 

F₂ = 1, 

F₃ = 1,

F₄ = 2,

…

F_n = F_n-1 + F_n-2

Series

Series is defined as the sum of all the elements in a sequence or it is defined as the sum of the sequence where the order of the terms does not matter. The series can be classified as finite or infinite depending on the types of sequence made of, whether it is finite or infinite. 

For example, 1+3+5+7+… is a series. The different types of series are:

• Geometric series
• Harmonic series
• Power series
• Alternating series
• Exponent series (P-series)

Difference-Sequence and Series

• A sequence is a specific format of elements in some definite order, whereas a series is the sum of the elements of the sequence. The elements of a  sequence follow a specific pattern whereas a series is the sum of elements in the sequence. 

A sequence is represented as x₁, x₂, x₃, x₄,…… x_n, whereas the series is represented as x₁+ x₂+ x₃+ x₄+……+ x_n-1 +x_n.

• The order of elements has to be maintained in a sequence i.e. in a sequence 7, 8, 9 is different from 7, 6, 5., whereas order is not important in the case of series 9 + 7 + 8 is the same as 7 + 6 + 5.
• The elements in the sequence follow a specific pattern whereas the series is the sum of elements in the sequence. 

Conclusion

Sequence and series are used to recognise patterns and it also has a contribution to the differential equation. A sequence is also known as the progression or collection of different elements in a defined order and a series is developed by sequence as it is the sum of all the components of the sequence. A sequence is represented as x₁+ x₂+ x₃+ x₄…… x_n, whereas the series can be represented as x₁+ x₂+ x₃+ x₄+……+ x_n-1 +x_n