Innings2
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Chapter 1: Patterns In Mathematics

    What is Mathematics?

    Patterns in Numbers

    Visualising Number Sequences

    Relations among Number Sequences

    Patterns in Shapes

    Relation to Number Sequences

    Summary

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Chapter 1: Patterns In Mathematics > Relations among Number Sequences

Relations among Number Sequences

Sometimes, number sequences can be related to each other in surprising ways.

Example: What happens when we start adding up odd numbers?

1 =

1 + =

1 + + =

1 + + + =

1 + + + + =

1 + + + + + = .....

This is a really beautiful pattern!

Can you see the pattern? Sum of 1 odd number is 1. Sum of 2 odd numbers is 22. Sum of 3 odd numbers is 32. The sum of ${a} odd numbers is ${a*a}.

Why does this happen? Do you think it will happen forever?

The answer is that the pattern does happen forever. But why?

As mentioned earlier, the reason why the pattern happens is just as important and exciting as the pattern itself.

A picture can explain it. Visualising with a picture can help explain the phenomenon. Recall that square numbers are made by counting the number of dots in a square grid.

How can we partition the dots in a square grid into odd numbers of dots: 1, 3, 5, 7,... ?

Think about it for a moment before reading further!

Here is how it can be done:

1 3 5 7 9 11 13

This picture now makes it evident that

1 + 3 + 5 + 7 + 9 + 11 = .

Because such a picture can be made for a square of any size, this explains why adding up odd numbers gives square numbers.

By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?

Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?

Another example of such a relation between sequences:

Adding up and down

Let us look at the following pattern:

1 = 1

1 + 2 + 1 =

1 + 2 + + 2 + 1 =

1 + 2 + 3 + + 3 + 2 + 1 =

1 + 2 + 3 + 4 + + 4 + 3 + 2 + 1 =

1 + 2 + 3 + 4 + 5 + + 5 + 4 + 3 + 2 + 1 =

This seems to be giving yet another way of getting the square numbers— by adding the counting numbers up and then down!

Can you find a similar pictorial explanation?

Counting numbers up and down

Watch how counting numbers form perfect squares when arranged in pictorially
Animation Speed:1.0x
Current Level
1
Pattern
1
Sum (Square)
1

(2) By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?

Instruction

The sequence counts up to and then mirrors back down to .
The middle number is 100, which means the total structure is .
From our previous explanation, such sequences always result in numbers.
From the pattern we identify: 1+2+3+...+(n−1)+n+(n−1)+...+3+2+1 = , where n is the number in the sequence.
For this case, n = , so the total sum is: =
1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1 = 10,000

(3) Which sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down?

Instruction

The "All 1’s" sequence is: ,,,,,,,…
Adding up: + + + = (upto 4 terms)
Adding up and down: + + + = (upto 4 terms)
The sum is in both cases.

(4) Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

Instruction

The counting numbers sequence is: , , , , ,,…
If we keep adding these numbers step by step, we get: 1, 1 + 2 = , 1 + 2 + 3 = , 1 + 2 + 3 + 4 = , 1 + 2 + 3 + 4 + 5 = ,…
This forms the numbers sequence: 1, 3, 6, 10, 15, 21, 28,…

(5) What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … Which sequence do you get? Why? Can you explain it with a picture?

Instruction

The triangular numbers sequence is: , , , , , , ,…
Now, let's sum consecutive terms: 1 + 3 = , 3 + 6 = , 6 + 10 =, 10 + 15 = , 15 + 21 =
This forms the numbers sequence: 4,9,16,25,36,… which is for n=2,3,4,5,6,….

(6) What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … ? Now add 1 to each of these numbers — what numbers do you get? Why does this happen?

Instruction

The powers of 2 sequence is: 1, 2, 4, , , , ,…
Now, let’s add them step by step: 1 + 2 = 3, 1 + 2 + 4 = , 1 + 2 + 4 + 8 = , 1 + 2 + 4 + 8 + 16 = . This gives the sequence: 1, 3, 7, 15, 31, ,,…
Now, let’s add 1 to each result: 1 + 1 = , 3 + 1 = , 7 + 1 =, 15 + 1 = , 31 + 1 = . This gives: 2, 4, 8, 16, 32, ,… which is exactly the powers of sequence!
The sum of the first n powers of 2 is one less than the next power of 2: 1 + 2 + 4 + 8 + ⋯ + 2n−1= 2n - 1
When you add 1, you get: 2n−1 + 1 =
This explains why the sums of powers of 2, when increased by 1, give back the powers of 2!

(7) What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?

Instruction

The triangular numbers sequence is: 1, 3, , , , ,, ,…
Multiply Each Triangular Number by 6: 1 × 6 = , 3 × 6 = , 6 × 6 = , 10 × 6 = , 15 × 6 =
Add 1 to Each Product: 6 + 1 = , 18 + 1 = , 36 + 1 = , 60 + 1 = , 90 + 1 =
7, 19, 37, 61, 91,… This sequence consists of prime numbers or numbers that are close to primes and follows a quadratic pattern.

(8) What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube?

Instruction

The sequence of hexagonal numbers is: 1, , , , ,, ,…
Adding Consecutive Hexagonal Numbers: 1, 1 + 7 = , 1 + 7 + 19 =, 1 + 7 + 19 + 37 = , 1 + 7 + 19 + 37 + 61 =
The resulting sequence is: 1, 8, 27, 64, 125,… which is the sequence of cubes: 13,23,33,43,53,.......
Each hexagonal number represents a layered structure that expands outward symmetrically. When we add them up, we essentially build a three-dimensional cube layer by layer, where each step increases the cube's side length by .

(9) Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?