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We can define a sequence as an arrangement of numbers in some definite order according to some rule.
Think of patterns that you see around you in daily life. Graphs, geometry, mandalas, snail shells, flower petals, and so on. All of these can be determined and represented mathematically.
This mathematical representation of such patterns is studied under sequence and series.
Any pattern when laid out in numbers and separated by commas is known as a sequence.
We can commonly represent sequences as x1,x2,x3,……xn, where 1,2,3 are the positions of the numbers and n is the nth term.
Whereas, series is defined as the sum of sequences, which means that if we add up the numbers of the sequence, then we get a series.
Example: 1+2+3+4+…..+n, where n is the nth term.
Series and sequence are the concepts that are often confused.
Suppose we have to find the sum of the arithmetic series 1,2,3,4 …100. We have to just put the values in the formula for the series.
Sequences and series are immensely useful when trying to do predictive or projective calculations. One simple example is score prediction, required run rate, projected score, etc. while watching a cricket match. All of these calculations are similar to studying numeric patterns and extending them or summing them up to visualise a future score, which is some steps further in the extension of the sequence that was observed from past scores.
Let us study the sequence and series formula.
Types of Sequence
There are mainly three types of sequences:
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Arithmetic Sequences
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Geometric Sequence
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Fibonacci Sequence
Arithmetic Sequence
Any sequence in which the difference between every successive term is constant is called Arithmetic Sequences.
This means that as we go further up in the sequence, the numbers keep increasing by an arbitrary constant value. And if we need to generate the next number, we simply add this arbitrary constant value again to the last number of the sequence and get a new number to extend the sequence.
Example:
3, 6, 9, 12, 15, 18, 21……..
+3 +3 +3 +3 +3 +3
Here, the difference between the two successive terms is 3. So, it is called the difference.
When we have to get the next number of this sequence, we simply add 3 to the last number of this sequence.
The difference is represented by “d”.
In the above example, we can see that a1 =3 and a2 = 6.
The difference between the two successive terms is
a2 – a1 = 3
a3 – a2 = 3
In an arithmetic sequence, if the first term is a1 and the common difference is d, then the nth term of the sequence is given by:
[a_{n} = a_{1} + (n-1)d]
Using the above formula, we can successfully determine any number of any given arithmetic sequence.
Geometric Sequences
A sequence in which every successive term has a constant ratio between them is called Geometric Sequence.
Constant ratio means that between every two numbers of the geometric sequence, there is an arbitrary constant, which is multiplied by the last number of the sequence to obtain the next number.
Example:
1, 4, 16, 64…
Here,
a1 =1
a2 = 4 = a1(4)
a3 = 16 = a2(4)
Here, we are multiplying it by 4 every time to get the next term. The ratio here is 4.
The ratio is denoted by “r”.
an = an-1⋅r or
an = a1⋅rn−1
[a_{n} = a_{n-1} times r]
Using the above formula, we can determine any number of any given geometric sequence.
Fibonacci Sequence
By adding the value of the two terms before the required term, we will get the next term. Such a type of sequence is called the Fibonacci sequence. There is no visible pattern.
The Fibonacci sequence is named after Leonardo Fibonacci, a famous Italian mathematician.
The Fibonacci sequence is very famous because it is the same pattern that is found in many natural wonders such as petals of flowers, shapes of eggs, etc. This sequence is also used to determine the golden ratio, which is a very important component in design and photography. The golden ratio is the ratio between any two numbers of the Fibonacci sequence. The rule of thirds in photography and graphic design is inspired by the golden ratio.
Example:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 88, 142 …
In the above sequence, we can see
a1 =0, a2 = 1
a3 = a2 + a1 = 0 + 1 =1
a4 = a3 + a2 = 1 + 1 =2, and so on.
So, the Fibonacci Sequence formula is
an = an-2 + an-1, n > 2
This is also called the Recursive Formula. Using this formula, we can calculate any number of the Fibonacci sequence.
Series
The summation of all the numbers of the sequence is called series. Generally, it is written as Sn.
Example:
If we have a sequence 1, 4, 7, 10, …
Then the series of this sequence is 1 + 4 + 7 + 10 +…
The Greek symbol sigma “Σ” is used for the series, which means “sum up”.
The series 4 + 8 + 12 + 16 + 20 + 24 can be expressed as [displaystylesumlimits_{i=1}^6 4n]
We read this expression as the sum of 4n as n ranges from 1 to 6.
For any sequence, its series can be calculated by summing up its numbers.
Types of Series
Having seen the types of sequences, let us move on to the types of series.
There are different types of series:
Arithmetic Series
If the sequence that we are calculating the series for is an arithmetic
An arithmetic series is the sum of a sequence ai, i = 1, 2,….n, in which each term is computed from the previous one by adding or subtracting a constant d.
If we are subtracting a constant, then it means that the common difference is a negative number.
Therefore, for i>1
ai = ai-1 + d = ai-2 + d=…………… =a1 + d(i-1),
where a is the first term and d is the difference between the terms, which is known as the common difference of the given series.
The Formula of Arithmetic Series
The formula for the nth term is given by an = a + (n – 1) d, where a is the first term, d is the difference, and n is the total number of the terms.
Let us now calculate the sum to n terms in an arithmetic series. The formula for the calculation is given below.
Sum of an Arithmetic Series
[S_{n} = frac{n}{2} 2a+(n-1)d]
Using the above formula, sum to the nth term can be found.
Geometric Series
Geometric series is the sum of all the terms of the geometric sequences, i.e., if the ratio between every term to its preceding term is always constant, then it is said to be a geometric series.
Therefore, when a geometric sequence is summed up, it is known as a geometric series.
We can calculate the sum to n terms of a geometric sequence using the below formula.
The Formula of Geometric Series
In general, we can define geometric series as
[displaystylesumlimits_{n=1}^infty ar^n = a + ar + ar^{2} + ar^{3} +……..ar^{n}]
Where a is the first term and r is the common ratio for the geometric series.
an = a1 r n – 1
Then the formula for the nth term is:
Sum of Geometric Series
[S_{n} = frac{a(1-rn)}{1-r}]
Geometric Progression (G.P.)
The sequence of numbers in which the next term of the sequence is obtained by multiplying or dividing the preceding number by the constant number is called a geometric progression. The constant number is called the common ratio. It is also known as Geometric Sequence.
a, ar, ar2, ar3, …, arn
In a geometric sequence, if the next number is obtained by dividing the previous number by a constant, then the common ratio is an inverse.
Arithmetic Mean
The arithmetic mean is the average of two numbers. If we have two numbers n and m, then we can include a number A in between these numbers so that the three numbers form an arithmetic sequence, like n, A, m.
In that case, the number A is the arithmetic mean of the numbers n and m.
Arithmetic Mean can be used to calculate the central tendency or the approximate centre point of an arithmetic sequence.
Arithmetic mean for an arithmetic sequence can be calculated using the below formula.
A = (n+m)/2
Geometric Mean
Geometric Mean is the average of two numbers in a geometric sequence. If p and q are the two numbers of the sequence, then the geometric mean will be
[GM = sqrt{pq}]
Geometric mean, similar to arithmetic mean, is used to calculate the central tendency or the approximate mid element of any given geometric sequence.
Harmonic Mean
By the harmonic mean definition, harmonic mean is the reciprocal of the arithmetic mean, the formula to define the harmonic mean “H” is given as follows:
Harmonic Mean(H) = [n/[(1/x_{1})+(1/x_{2})+(1/x_{3})……(1/x_{n})]]
Where
n is the total number of terms and
x1, x2, x3,…, xn are the individual values up to nth terms.
Solved Examples
Example 1: What will be the 6th number of the sequence if the 5th term is 12 and the 7th term is 24?
Solution: As the two numbers are given, the 6th number will be the arithmetic mean of the two given numbers.
AM = 12 + 24 / 2
= 36/2
= 18
Hence, the 6th term will be 18.
Example 2: Find the geometric mean of 2 and 18.
Solution: Formula to calculate the geometric mean.
p = 2 and q = 18
GM = [sqrt{pq}]
= [sqrt{2times18}]
= [sqrt{36}]
= 6
Quiz Time
1. What is the ninth term of the geometric sequence, 3, 6, 12, 24, …?
Sol: Given the first term is 3.
Common ration=63=2
So, ninth term a9=ar8=328
=3356
=768.
3. What is the sum of the first ten terms of the geometric sequence 5, 15, 45, …?
Sol: From a given series, we can determine that the first term is 5.
Common ratio=153=5
Formula to find sum of n terms = a1(rn-1)r-1
=5(310-1)3-1
=5590482
=147620