Patterns in Whole Numbers: Definitions, Facts, and Examples

A pattern is a set of shapes, numbers or letters that can be repeated according to a rule so that a sequence is formed. Patterns can be found all around us. Artists, architects and designers use the idea of symmetry and patterns in their profession. Rangoli, which is very popular in our country, is also made with the concept of pattern.

Patterns can be of various types – increasing, decreasing, logical, skipping, changing directions, a combination of these, etc. Patterns can be in whole numbers too! In this article, we will learn to make and frame the pattern in whole numbers.

What are Patterns?

Nature is full of patterns. In flowers, petals follow a pattern. In the branch of a tree, leaves follow a pattern. A pattern is formed when shapes, letters, or alphabets are repeated following a rule. Thus, a pattern is an arrangement of numbers, shapes, etc. By observing the patterns, we recognize and extend them. Below are given some examples of patterns using numbers and shapes.

Example 1: \(Aa\)         \(Bb\)           \(Cc\)             \(Dd\)              \(Ee\)

Rule: Capital letters are followed by small letters in the pattern.

Example 2: \(@\)           \(@@\)               \(@@@\)              \(@@@@\)                \(@@@@@\)

Rule: One \(@\) is added each time.

Example 3: \(10\)          \(8\)             \(6\)              \(4\)            \(2\)

Rule: In the pattern, \(2\) is subtracted from the previous number.

Example 4: \(3\)            \(6\)             \(9\)             \(12\)         \(15\)

Rule: In the pattern, \(3\) is added to the previous number.

Example 5: \(2V2\)            \(4V4\)               \(6V6\)            \(8V8\)             \(10V10\)

Rule: There is addition as well as repetition in the pattern. Multiples of \(2\) are placed before and after \(V\) in this pattern.

What is Number Pattern?

Numbers are fascinating and astounding because they contain many beautiful patterns and sequences that are incredibly fascinating. A list of numbers with a common trait is called a pattern of numbers. In math, solving problems with number patterns increases a student’s logical thinking and mathematical reasoning ability.

To answer any inquiry about whole number patterns in a sequence, the rule used to create the pattern is to be initially understood.

Patterns in Whole Numbers

Let us recall an exciting way of arranging numbers in a square and a triangle using small dots. Some numbers like \(3, 6,\) and \(10\) can be arranged in a triangle. Such numbers are called triangular numbers.

Some numbers like \(4, 9,\) and \(16\) can be arranged in a square. Such numbers are called square numbers.

Now, let us look at other shapes in which you can arrange numbers. You can arrange all numbers in a line.

You can arrange some numbers in a rectangle. \(12\) can be shown in \(2\) different rectangles.

Let us look at some more patterns and their explanation.

Pattern 1: 

\(1 + 3 = 4 = 2 \times 2 = {2^2}\)

\(1 + 3 + 5 = 9 = 3 \times 3 = {3^2}\)

\(1 + 3 + 5 + 7 = 16 = 4 \times 4 = {4^2}\)

The following \(3\) steps of this pattern will be:

\(1 + 3 + 5 + 7 + 9 = 25 = 5 \times 5 = {5^2}\)

\(1 + 3 + 5 + 7 + 9 + 11 = 36 = 6 \times 6 = {6^2}\)

\(1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 7 \times 7 = {7^2}\)

According to this pattern, the sum of the first n odd numbers\( = {n^2}.\)

That is, \(1+3+5+7+9+11+……\)up to \(n\) odd numbers\( = {n^2}.\)

In the same way,

1. The sum of first n even numbers is \(n(n+1).\)

\(⇒2+4+6+8+10=5(5+1)=30\)

\( \Rightarrow 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 9(9 + 1) = 90\)

2. The sum of first \(n\) numbers is \(\frac{{n(n + 1)}}{2}\)

\( \Rightarrow 1 + 2 + 3 + 4 + 5 + 6 = \frac{{6(6 + 1)}}{2} = 21\)

\( \Rightarrow 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = \frac{{10(10 + 1)}}{2} = 55\)

Pattern 2: 

\(1+2+3+4+5=15\)

\(2+3+4+5+6=20\)

\(3+4+5+6+7=25\)

The next two steps of this pattern will be: 

\(4+5+6+7+8=30\)

\(5+6+7+8+9=35\)

Pattern 3: 

\(1×1=1\)

\(11×11=121\)

\(111×111=12321\)

The following \(3\) steps of this pattern will be

\(1111×1111=1234321\)

\(11111×11111=123454321\)

\(111111×111111=12345654321\)

Number Pattern Types

As we have already learnt about the patterns and the number patterns above, let us learn about the number pattern types. There can be various number pattern types, like square number patterns, cube number patterns, triangular number patterns, etc.

Square Number Pattern

What do you notice about the first \(6\) square numbers given below?? They are coloured in a certain way to help you discover one of the patterns and properties. Look at the below-given picture very carefully.

What is the pattern that you identified????

You can see that:

\(1×1=1=1=1\)

\(2×2=4=1+3=1+3\)

\(3×3=9=4+5=1+3+5\)

\(4×4=16=9+7=1+3+5+7\)

\(5×6=16=16+9=1+3+5+7+9\)

\(6×6=36=25+11=1+3+5+7+9+11\)

This is true for any square number \(x.\)

Thus, \({x^2}\) is the sum of the first \(x\) odd numbers.

Cube Number Pattern

Let us look at sums of cubes numbers.

The first cube number is \(1,\) which is \({1^3}\) that is also equal to \({1^2}.\)

The sum of the first two cube numbers\( = {1^3} + {2^3} = 1 + 8 = 9 = {3^2}\)

The sum of the first three cube numbers\( = {1^3} + {2^3} + {3^3} = 1 + 8 + 27 = 36 = {6^2}\)

The sum of the first four cube numbers\( = {1^3} + {2^3} + {3^3} + {4^3} = 1 + 8 + 27 + 64 = 100 = {10^2}\)

The sum of the first five cube numbers\( = {1^3} + {2^3} + {3^3} + {4^3} + {5^3} = 1 + 8 + 27 + 64 + 125 = 225 = {15^2}\)

What pattern do you notice???

We have already learnt the unique name for the numbers like \(1, 3, 6, 10\) and \(15.\) They are known as triangular numbers.

Thus, we can say that the sum of the first \(x\) cube numbers is equal to the square of the \({x^{{\rm{th}}}}\) triangular number. 

This can be shown with the help of the following picture.

Thus, cube numbers can also be written as the sum of consecutive odd numbers.

Using Number Patterns to Solve Problems

We can also use the number patterns to simplify mathematical problems. Observe the below-given patterns.

1. \(64+9=64+10-1\)

\(=(64+10)-1\)

\(=74-1=73\)

2. \(64+19=64+20-1\)

\(=(64+20)-1\)

\(=84-1=73\)

3. \(134-8=134-10+2\)

\(=(134-10)+2\)

\(=124+2=126\)

4. \(134-18=134-20+2\)

\(=(134-20)+2\)

\(=114+2=116\)

Thus it becomes simple to calculate the sum or the difference of two numbers when one of the numbers is closer to \(10, 20, 30, 40, 50.\)

Solved Examples – Patterns in Whole Numbers

Q.1. Observe the number pattern and fill up the blanks.
\(5×11=55\)
\(55×101=5555\)
\(555×1001=555555\)
\(5555 \times 1001 = \_\_\_\_\_\_\_\)
\(55555 \times \_\_\_\_\_\_ = \_\_\_\_\_\_\)
Ans: From the pattern, we can observe that there is an amendment of \(5\) in the first numbers, and then a \(0\) is included in between the second number, and subsequently, two \(5\) are being added in the answer.
Hence, the complete pattern will become as
\(5×11=55\)
\(55×101=5555\)
\(555×1001=555555\)
\(5555×1001=55555555\)
\(55555×10001=5555555555\)

Q.2. Study the given pattern of triangular numbers and extend it to \(3\) more steps.
\(8×1+1=9=3×3\)
\(8×3+1=25=5×5\)
\(8×6+1=49=7×7\)
\(8×10+1=81=9×9\)
Ans: We know that the numbers like \(3, 6, 10, 15,\) and so on can be arranged in the shape of a triangle using dots, and thus, such numbers are known as triangular numbers.
By observing the given pattern, the following \(3\) steps will be as follows,
\(8×15+1=121=11×11\)
\(8×21+1=169=13×13\)
\(8×28+1=225=15×15\)
Hence, the complete pattern is shown as below;
\(8×1+1=9=3×3\)
\(8×3+1=25=5×5\)
\(8×6+1=49=7×7\)
\(8×10+1=81=9×9\)
\(8×15+1=121=11×11\)
\(8×21+1=169=13×13\)
\(8×28+1=225=15×15\)

Q.3. Observe the number pattern and write the next \(3\) steps.
\(1×8+1=9\)
\(12×8+2=98\)
\(123×8+3=987\)
\(1234×8+4=9876\)
Ans: By observing the given pattern, we can write the following \(3\) steps as;
\(12345×8+5=98765\)
\(123456×8+6=987654\)
\(1234567×8+7=9876543\)
Hence, the complete pattern is shown as follows;
\(1×8+1=9\)
\(12×8+2=98\)
\(123×8+3=987\)
\(1234×8+4=9876\)
\(12345×8+5=98765\)
\(123456×8+6=987654\)
\(1234567×8+7=9876543\)

Q.4. For the pattern given below, write the next \(3\) steps:
\(9×9+7=88\)
\(98×9+6=888\)
\(987×9+5=8888\)
Ans: The given pattern is as follows;
\(9×9+7=88\)
\(98×9+6=888\)
\(987×9+5=8888\)
By observing the pattern, the next three steps of the pattern will be
\(9876×9+4=88888\)
\(98765×9+3=888888\)
\(987654×9+2=8888888\)
Hence, the complete pattern is as follows;
\(9×9+7=88\)
\(98×9+6=888\)
\(987×9+5=8888\)
\(9876×9+4=88888\)
\(98765×9+3=888888\)
\(987654×9+2=8888888\)

Q.5. For the pattern given below, write the next \(3\) steps.
\(111÷3=37\)
\(222÷6=37\)
\(333÷9=37\)
Ans: From the given pattern, we can deduce that the dividend remains a three-digit number only, but the next number in the pattern is the following natural number than the previous one. Also, the divisor is a multiple of \(3.\) So the next \(3\) steps will be as follows.
\(444÷12=37\)
\(555÷15=37\)
\(666÷18=37\)
Hence, the complete pattern is as follows;
\(111÷3=37\)
\(222÷6=37\)
\(333÷9=37\)
\(444÷12=37\)
\(555÷15=37\)
\(666÷18=37\)

Summary

In this article, we first had a quick view of patterns and learnt them well with the help of illustrations. Later we learnt the definition of number patterns and then patterns in whole numbers in detail. We also learnt the types of patterns in numbers like square number patterns, cube number patterns, triangular number patterns, etc.

And, lastly, we solved some examples based on the number pattern on the whole number to strengthen our grip over the concept.

Frequently Asked Questions (FAQs)

Q.1. What are the number patterns?
Ans: A pattern is a set of shapes, numbers, or letters that can be repeated according to a rule to form a sequence. Numbers are fascinating and astounding because they contain many beautiful patterns and sequences that are fascinating. A list of numbers with a common trait is called a pattern of numbers.

Q.2. Are there patterns in numbers?
Ans: Yes, there are patterns in numbers as well.

Q.3. What is an example of a number pattern?
Ans: An example of a number pattern is shown below.
\(1+3+5+7+9+11+13+15\) is the sum of the first eight odd natural numbers.
Another example that can be added is the sum of first \(x\) natural numbers us \(x\left( {\frac{{x + 1}}{2}} \right)\)
Therefore, the sum of the first \(11\) natural numbers is
\(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = \frac{{11(11 + 1)}}{2} = 66\)

Q.4. What is it called when there is a pattern in numbers?
Ans: When there is a pattern in numbers, we call it a number pattern.

Q.5. How do you identify patterns in numbers?
Ans: To identify the pattern, the easiest way is to compare the first and the next preceding line to see the change and then compare the third line with the second line, and then we can find the patterns in number. Also, it can help us to find the next line in the series.