TYPES OF SEQUENCES AND SERIES

Understanding sequences and series is a crucial part of studying mathematics as it is applicable in various fields such as computer programming, finance, statistics and physics. A sequence, also called a progression, is defined as the arrangement of individual terms in an orderly manner. These individual terms of a sequence, when added together, give rise to a series. Therefore, a series is the sum of terms in a progression. The order of elements or terms is important in a sequence but not in a series. These sequences and series can be classified into arithmetic progression, geometric progression, harmonic progression and Fibonacci progression.

SEQUENCES AND SERIES:

Sequences can be classified as finite or infinite. A finite sequence is where the last term of the sequence is known, whereas an infinite progression is a never ending progression. Take the following progression as an example:

2,4,6,8,10……n

Each number is obtained in this progression of numbers by adding the number 2 to the preceding term. Therefore, the sequence does not have a definite number of terms and is called an infinite series.

Depending upon the set of rules followed by the sequences and series, they may be classified into four types:

ARITHMETIC PROGRESSION:

A progression or sequence in which each element or term is derived by addition or subtraction of a fixed number from the preceding term is called an arithmetic progression or an A.P. An and can be represented as

a, a+ d, a+ 2d, a+ 3d, …. a+ nd

where a: first term in the sequence

        d: common difference

        n: the number of terms

The nth term in an arithmetic progression can be calculated by the formula,

an = a+ (n−1)d 

The series for a finite arithmetic progression can be calculated by the formula,

Sn = n/2 (2a + (n−1)d)

where Sn: Sum of first n terms in the sequence

Let’s take an example. Calculate the sum of the first 20 terms and calculate the 10th term for the given A.P. 4, 9, 11, 16, 21…. n.

For sum of first 20 terms,

Sn = n/2 (2a + (n−1)d)

Here, n = 20

        a = 4

        d = 5

Therefore, Sn = n/2 (2a+ (n−1)d)

S20 = 20/2 (2 x 4 + (20−1)5)

 = 1030

For the 10th term,

an = a1+ (n−1)d 

a10 = 4+ (10−1)5 

= 49

GEOMETRIC PROGRESSION:

A progression or sequence in which each element or term is derived by multiplication or division of a constant ratio/number from the preceding term is called a geometric progression or G.P. and can be represented as,

a , ar , ar2 , ar3 , …arn – 1

where a: first term in the progression

        r: common ratio

        n: the number of terms

The nth term in a geometric progression can be calculated by the formula,

nth term = a rn-1

The series for a geometric progression finite can be calculated by the formula,

Sn = a (1 – rn) / (1 – r).

Let’s take an example. Calculate the sum of the first 10 terms for the given G.P. 5, 10, 20, 40, 80… and so on.

Sn = a (1 – rn) / (1 – r),

 S10 = 5 (1 – 210) / (1 – 2)

= 5115

HARMONIC PROGRESSION:

A progression or sequence in which each element or term is derived from the reciprocal of the corresponding term in an arithmetic progression is called a harmonic progression or H.P. and is represented as,

1/a1, 1/a2, 1/a3, 1/a4, … 1/an                             

Where a1, a2, a3, a4, … an is the arithmetic progression.

The nth term of the harmonic progression can be calculated by the formula,

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

The series for a finite harmonic progression can be calculated by the formula,

Sn= 1/d (ln(2a + (2n – 1)d) / (2a – d))

Where ln is the natural logarithm.

An example of the harmonic progression is as follows:

1/2, 1/4, 1/6, 1/8, 1/10, …. 1/2n.

FIBONACCI SEQUENCE:

A Fibonacci sequence is the sequence of numbers in which each term is the sum of two preceding terms, starting with 0 and 1 as the first two terms. Therefore, it can be represented as,

0, 1, 1, 2, 3, 5, 8, 13…n.

Here, F0 = 0, F1 = 1, F2 = 1, Fn being the nth term in the Fibonacci series.

The nth term in the Fibonacci series can be calculated by the formula,

Fn = Fn-1 + Fn-2