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Introduction
In this blog we will learn about sequence and series and some terminologies related to its sequence and series. Finally, we will also look at the difference between sequence and series. But before we look into different types of sequence and series, we should first look at the basic definition of sequence and series.
One of the fundamental concepts in Arithmetic is sequence and series.
What is a Sequence?
A sequence is sometimes referred to as a progression, creating a series. Sequences are the ordered groupings of numbers that follow certain criteria.
What is Series?
Series is the sum of the parts in the sequence. For example, 1, 2, 3, 4, 5 is a four-element sequence, and the equivalent series is 1 + 2 + 3 + 4 + 5, with the sum or value of the series being 15.
When doing predictive or projective computations, sequences and series are pretty beneficial. While watching a cricket match, we often see score prediction, needed run rate, anticipated score, etc. These are practical applications of sequence and series. These computations are similar to studying numerical patterns and extending or summing them up to visualize a future score, which is a few steps further in the extension of the sequence observed from previous scores.
Difference b/w Sequence and Series
Types Of Sequence and Series
There are various types of sequence and series and here we will discuss some of the most used sequences and series:
Arithmetic Sequence
Geometric Sequence
Harmonic Sequence
Fibonacci Sequence
Arithmetic Sequence
Arithmetic Sequence, commonly known as AP (Arithmetic Progression), are sequences in which the difference between each subsequent term is constant.
This means that as we progress through the sequence, the numbers continue to increase by an arbitrary constant value. And if we need to produce the following number, we add this random constant value to the last number in the sequence to obtain a new number to extend the sequence.
Arithmetic Series Formulas:
For example, consider an arithmetic series:
1+3+5+7+9…
Here we have to find the last term, and we need to calculate the sum of the series. There are a total of 20 terms in the series.
As you can see the first term i.e, a is equal to 1,
and the common difference i.e, d is (3-1) i.e, 2
a1 = 1 = 1 + 2*0
a2 = 3 = a1 + 2*1
a3 = 5 = a1 + 2*2
Since an = a + (n-1)d,
an = 1 + (20-1)*2 = 39
Therefore the last term is 39.
Sum = n/2 (2*a + (n-1)*d) = 20/2 (2*1 + (20-1)*2) = 10 (2 + 38) = 400
Geometric Sequence
Geometric sequence, commonly known as GP (Geometric Progression) is a sequence in which each succeeding phrase has a constant ratio between them.
The constant ratio is an arbitrary constant between every two numbers in the geometric series which is multiplied by the last number in the sequence to get the following number.
Geometric Series Formulas:
For example, consider a geometric series:
2+4+8+16+32 …
Here we have to find the last term, and we also need to calculate the sum of the series. There are a total of 10 terms in the series.
As you can see the first term i.e, a is equal to 2,
and the common ratio i.e, r is (4/2) i.e, 2
a1 = 2
a2 = a1*r = 2*2 = 4
a3 = a1*r² = 2*2*2 = 8
Since an = ar⁽ⁿ⁻¹⁾,
an = 2*2⁽¹⁰⁻¹⁾ = 2¹⁰
Therefore the last term is 1024.
Since this series is a finite series we will use the finite series formula to calculate the sum.
Sum = a(1-rⁿ)/(1-r) = a(rⁿ-1)/(r-1) = 2(2¹⁰-1)/(2-1) = 2(1024-1) = 2046
Harmonic Sequence
Harmonic sequence, commonly known as HP (Harmonic Progression), is a real-number sequence determined by calculating the arithmetic progression's reciprocals that do not contain 0. Any phrase in the sequence is assumed to be the harmonic mean of its two neighbors in harmonic progression. The sequence a, b, c, d,..., for example, is regarded as an arithmetic progression; the harmonic progression may be represented as 1/a, 1/b, 1/c, 1/d,...
Harmonic Series Formulas:
For example, consider a harmonic series:
5+5/2+5/3+…
Here we have to find the 6th term of the series.
A.P. = 1/5, 2/5, 3/5,
Here, common difference = a2-a1= a3-a2 = 1/5
Now, we need to find the 6th term
a₆ = 1/5 + (6-1)*1/5 = 1/5 + 1 = 6/5
Since HP is reciprocal of AP, the 6th term of HP would be 5/6.
Fibonacci Sequence
A Fibonacci sequence is a sequence where we get the next term by adding the previous two terms. The Fibonacci sequence is named after the famous Italian mathematician Leonardo Fibonacci.
The Fibonacci sequence is well-known since it is the same pattern seen in many natural wonders such as flower petals, egg forms, etc. This sequence is also used to calculate the golden ratio, a critical component in design and photography. The golden ratio is the ratio of any two Fibonacci sequence numbers. The golden ratio inspired the rule of thirds in photography and graphic design.
For Example, consider a fibonacci sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 88, 142, 230 …
In the above sequence, we can see
a1 = 1, a2 = 1
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 1 + 2 = 3, and so on.
So, the Fibonacci Sequence formula is
an = an-2 + an-1, n > 2
What is an arithmetic mean? The arithmetic mean is the average of two numbers in an arithmetic sequence. Let us suppose we have to number n and m, and we put a number A between them to make an arithmetic sequence, such as n, A, m. Formula for calculating arithmetic mean: AM = (n+m)/2
What is the geometric mean of two numbers? Geometric Mean is the average of two numbers in a geometric sequence. If p and q are the sequence's two numbers, then the geometric mean is GM = √pq
What are real-life applications of sequence and series? Arithmetic sequences are used in everyday life for various reasons, including assessing the capacity of an auditorium, calculating predicted revenues from working for a firm, and constructing wood heaps using stacks of logs. Arithmetic sequences are algebra and geometry tools that aid mathematicians and others in problem-solving.
Key Takeaways
This article discussed sequence and series and some basic terminologies related to them. I hope you have gained some insight into this topic of sequence and series.
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