In statistics and mathematics, there are two corresponding terms that we usually listen to series and sequences. Both of these terms are interrelated with each other. If we look at history, in the 17th century Carl Friedrich Gauss was the first mathematician who gave the idea of performing long calculations.
After that many mathematicians worked on this idea and derived different formulas and identities. Generally, there are three types of sequences arithmetic sequence, geometric sequence, and harmonic sequence. The details are given below. A sequence is also known as progression.
In this article, we will study the basic definition of the term “sequence”. We will also explain the types of sequences arithmetic and geometric. There is an example section where you can easily understand the working of these terms.
Definition of Sequence
“A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.”
The sequence contains elements just like the set; the total number of elements is known as the length of the sequence.
Types of the Sequence
As we have discussed in the introductory paragraph that there are three types of sequence, but in this article, we will just discuss two types of it.
- Arithmetic sequence.
- Geometric sequence.
Arithmetic Sequence
A sequence of terms whose gap is constant between the terms is called an arithmetic sequence. In this type of sequence, we can obtain the sequence’s next term by adding up the constant number to the previous term.
The gap between the terms is known as the common difference and is denoted by “d”. We can easily make the sequence when the 1st term and the common difference are given. In set theory, the arithmetic sequence is widely used.
The sequence of odd numbers, whole numbers, and even numbers are the best examples of arithmetic sequence because the common difference between the terms of these sets is the same.
Formulas of Arithmetic Sequence
Generally, the arithmetic sequence is categorized into three sub-categories.
- Common difference
- Nth term
- Sum of the sequence
The formulas for these three sub-categories are as follows.
Common Difference
If the common difference is not given, then we can find it using the following formula.
d = xn – xn-1
Nth term:
If you want to find the nth term of the sequence, you can use the formula given below.
Nth term = xn = x1 + (n – 1) d
The nth is represented by xn
Sum of the Sequence
There are two different formulas for finding the sum of an arithmetic sequence.
- When the last term is not given.
Sum = S = n/2 * (2x1 + (n – 1) d)
- When the last term is given.
Sum = S = n/2 (x1 + xn)
- “S” is the sum of the sequence.
- “n” is the total number of terms
- “x1” is the first term.
- “xn” is the last term of the sequence.
- “d” is the common difference
Geometric Sequence
The geometric sequence is a sequence of numbers in which each term after the first can be obtained by multiplying the previous one by a fixed number. This non-zero number is known as the common ratio. The common ratio is denoted by “r”.
Formulas of Geometric Sequence
Generally, the geometric sequence is categorized into three sub-categories.
- Common ratio
- Nth term
- Sum of the sequence
Common ratio:
The geometric sequence’s common ratio can be obtained by using the following formula.
Common ratio = r = xn / xn-1
Nth term:
If you want to find the nth term of the sequence, you can use the formula given below.
Nth term = xn = arn-1
The nth of the sequence is denoted by xn.
Sum of the sequence:
The geometric sequence of finite terms can be obtained by the following formula.
- If the common ratio “r” is positive and is greater than 1. (r is not equal to 1)
Sum = S = a [(rn – 1) / (r – 1)]
- If the common ratio “r” is less than 1.
Sum = S = a [(1 – rn) / (1 – r)]
Examples
Example 1: (For arithmetic progression)
Find the 30th term of the given arithmetic sequence, 4, 7, 10, 13, 16, 19, 22…
Solution
Step I: Write the given data.
4, 7, 10, 13, 16, 19, 22
Step II: Find the common difference.
x1 = 4
x2 = 7
d = x2 – x1
d = 7 – 4
d = 3
Step III: Find the nth term of the sequence.
Nth term = xn = x1 + (n – 1) d
Nth term = xn = 4 + (n – 1)3
Nth term = xn = 5 + 3n – 3
Nth term = xn = 3n – 2
Step IV: Put n = 50, because we have to find the 30th term.
30th term = x30 = 3(30) – 2
30th term = x30 = 90 – 2
30th term = x30 = 88
An arithmetic sequence calculator by Allmath can help you to solve these lengthy calculations within an instant.
Example 2: (For geometric sequence).
Find the next consecutive four terms of a geometric sequence whose first term is 2, where r = 3.
Solution:
Step 1: Write the given data.
First term = a = 2
Common ratio = r = 3
Step 2: Select the suitable formula
We will use the nth-term formula to find the first five terms of the geometric sequence.
xn = arn-1
For the second term (n = 2):
x2 = 2(3)2 – 1 ∴ a = 2, r = 3, and n = 2
x2 = 2(3)
x2 = 6
For the third term (n = 3):
x3 = 2(3)3 – 1 ∴ a = 2, r = 3, and n = 3
x3 = 2(3)2
x3 = 2(9)
x3 = 18
For the fourth term (n = 4):
x4 = 2(3)4-1 ∴ a = 2, r = 3, and n = 4
x4 = 2(3)3
x4 = 2(27)
x4 = 54
For the fifth term (n = 4):
x5 = 2(3) 5 – 1 ∴ a = 2, r = 3, and n = 5
x5 = 2(3) 4
x5 = 2(81)
x5 = 162
The terms are 6, 18, 54, and 162.
Final Thoughts
In this article, we have studied the basic definition of the sequence. We have also studied the types and the theory related to them and in the example section, we have studied the working principle of the arithmetic and geometric progression.
You have noted that these terms are not quite difficult, and now you can easily solve all the problems related to AP and GP.
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