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Sequences are lists of numbers in a certain order, such as 2,5,8. Some sequences have a pattern that can be used to make them go on forever. For example, 2,5,8 follows the pattern “add 3,” so we can keep going with the pattern. There can be formulas for sequences that tell us how to find any given term in the sequence.
The Cube Numbers
A cube number is the result of multiplying three identical numbers together. In other words, a cube number is what you get when you multiply a number by itself and then by itself again.
1, 8, 27, 64,125,216,343,512,729, …
They are the cubes of the counting numbers (which start at 1):
1 (=1×1×1)
8 (=2×2×2)
27 (=3×3×3)
64 (=4×4×4)
etc…
Types of Sequence and Series
Some common examples of sequences include:
- Arithmetic Sequences
- Geometric Sequences
- Triangular series
- Square series
- Cube series
- Fibonacci Numbers
- Twin series
Arithmetic Sequences
This is a type of number sequence where the next term is found by adding a constant value to the previous term. When the first term is x1 and d is the difference between two consecutive terms, the sequence is described by the following formula:
xn = x1 + (n-1) d
where;
The nth term is xn.
The first term is x1, the number of terms is n, and the difference between any two terms is d.
Geometric Sequences
The geometric series is a set of numbers where the next number is found by multiplying the number before it by a constant called the common ratio. The formula sums up the geometric number series as a whole:
xn = x1 × rn-1
where;
nth term = xn,
X1 is the first term,
r is the average ratio, and
n is the amount of terms.
Triangular Series
This is a series of numbers where the first number stands for the terms linked to the dots in the picture. For a number with three sides, the dot shows how many dots are needed to fill a triangle. The formula for the triangle number series is;
x n = (n2 + n) / 2.
Square Series
When you multiply an integer by itself, you get a square number. The formula shows a square number of series. Square numbers are always positive.
xn = n2
Cube Series
One way to make a cube number series is to multiply a number by itself three times. For cube number series in general, the formula is:
xn = n3.
Fibonacci Numbers
The Fibonacci numbers are an interesting series of numbers that start with 0 and 1. Each number is made by adding the two numbers before it. The definition of sequence is that F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2.
Twin Series
By definition, a twin number series is made up of two series put together. When the terms of a twin series change, they can create a third, separate series.
The numbers 3, 4, 8, 10.13, 16, etc. are an example of a twin series.
By looking at this series closely, we can make two other series: 1, 3, 8, 13, and 2, 4, 10, 16.
Example of Number Sequence
Example 1
It’s important to remember that the common difference isn’t always a positive number. There can be a negative common difference, as shown by the following series of numbers:
25, 23, 21, 19, 17, 15…….
In this case, the difference between them is -2. We can find any term in the series by using the math formula. To get, say, the fourth term.
4th term = 25 + (4-1) – 2
=25 – 6
=19
Example 2
Take the numbers 3, 8, 13, 18, 23, and 28 as an example.
8 – 3 = 5 is a common way to find the difference;
“3” is the first term. For example, to find the fifth term using the math formula, change the first term’s value to 3, the common difference to 5, and n to 5.
The 5th term is equal to 3 + (5-1) + 5.
=23
Example 3
As the following example shows, a geometric series can have terms that go down.
2187, 729, 243, 81,
In this case, you can find the common ratio by dividing the previous term by the next term. This set has a ratio of 3 in common.
Example 4
Take a look at the following series of triangles:
1, 3, 6, 10, 15, 21………….
This pattern is made up of dots that fit together to make a triangle. You can make a sequence by putting more dots in another row and counting them all.
Example 5
A Fibonacci number series example is:
0, 1, 1, 2, 3, 5, 8, 13, …
For example, 0+1+1=2 is used to figure out the third term of this series. In the same way, 8 + 5 = 13 is used to figure out the 7th term.
Conclusion
In short, when solving problems with number sequences and patterns, you need to look at how these numbers are related to each other. You should look for a mathematical connection, such as a relationship between subtraction and addition. To find their common ratio, divide and multiply the terms to see if they have any geometric relationships.
