Sequences and Series: Definition

A sequence is a collection of different terms or elements or objects that have been arranged in a specific order. The sum of all the terms in a sequence can be used to generalize a series. An individual term in a sequence might appear multiple times in different positions. A sequence’s length is equal to the number of terms in it, and it might be finite or infinite. Let us now define the terms.

Sequence 

It is defined as a group of integers arranged in a particular pattern, with each number in the sequence being referred to as a term.

For instance, 8, 12, 18, 22, 28,… is a series in which the ellipsis sign at the end denotes that the list continues indefinitely. The first term in this series is 8, the second term is 12 and so on. There is a difference between each component in the series, known as the common difference. The common difference in the example above is 5. 

Sequences include the following

Arithmetic Sequence

An arithmetic sequence is a sequence in which each term is obtained by adding or subtracting a defined number known as the common difference from the previous number. It can also be stated as a sequence of numbers that are ordered and have a common difference between each term. In an arithmetic sequence, the difference between consecutive terms is always equal. An arithmetic sequence is one in which we add or subtract by the same number each time.

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

For example- 2, 12 , 22, 32, 42, 52…

Here, we observe that the common difference is 3. 

For finding the nth term we use the formula-

an = a1 + (n − 1)d 

where d is a common difference.

Geometric Sequence

A geometric sequence is a sequence in which each of the components in the series is formed by multiplying or dividing a constant number with the prior one. A GP’s common ratio is calculated by dividing the ratio between any two consecutive terms in the sequence by the prior term. It’s a special kind of sequence. It’s a sequence in which each phrase (except the first) is multiplied by a fixed number to generate the following term. To determine the preceding word in the geometric sequence, simply divide by the same common ratio.

The GP is of form a, ar, ar2…, where ‘a’ is the first term and r is the series’ common ratio. A positive or negative real number can be used as the common ratio.

The sum of first n terms in the GP can be calculated by-

sn = a(1 – rn)/(1 – r)       when, r < 1

sn = a(rn -1)/(r – 1)         when, r > 1

Harmonic Sequence

In mathematics, a harmonic sequence is a set of numbers a1, a2, a3,… whose reciprocals 1/a1, 1/a2, 1/a3,… form an AP (no. segregated by a common difference).

A series is defined as the sum of a sequence, and the harmonic series is an example of an infinite series that has no limit. In other words, the partial sums generated by adding successive terms rise indefinitely, or the sum approaches to infinity.

The nth term of the HP 

an = 1/ [a1 + (n-1) d]

where d=common difference

Fibonacci Sequence

The Fibonacci sequence is a set of integers in which each subsequent number is the sum of the two preceding numbers. The series begins at 0 and 1 and continues indefinitely: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. A mathematical equation can be used to describe the Fibonacci sequence where xn is the nth term

 xn+2 = Xn+1 + Xn

Series

A series is defined as the sum of all members in a sequence, or as the sum of a sequence where the order of the words is irrelevant. Depending on the types of sequence used and whether it is finite or infinite, the series can be classed as finite or infinite.

1+4+7+10+… is an example of a series.

The following are the various categories of series:

• Exponent series 

• Geometric series 

• Harmonic series 

• Power series 

• Alternating series (P-series)

Conclusion

A sequence is an itemized collection of elements that allows for any type of repetition, whereas a series is the summation of all elements. One of the instances of sequence and series is an arithmetic progression. By adding all the terms of a sequence we obtain a series. Therefore the series is the summation of all the elements in a sequence.