Types of Sequence and Series, Formula and Solved Example

Mathematics is all about sequences and series. A sequence is a list of elements that allows for any number of repetitions, whereas a series is the sum of all elements. For instance, 2, 4, 6, 8, 10 is a five-element sequence, and the corresponding series is 2+4+6+8+10, with the sum of the series being 30. Let us now comprehend them. One example of sequence and series is an AP (arithmetic progression).

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

Sequences of Various Types

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)

Conclusion

A sequence is a collection of numbers arranged in a specific order according to a set of rules. Make a note of the patterns you notice in your daily life. You can draw graphs, geometry, mandalas, snail shells, flower petals, and other things. In some way, all of these things can be calculated and expressed numerically. Under the headings of sequence and sequences, this mathematical description of such patterns is investigated further. Any pattern that is set out in numbers and separated by commas is referred to as a sequence.