Squares as Diamond Dot Patterns
Count the dots row by row to see the square.
Click to zoom
Count the number of dots in each row and add them together. This sum results in a square number:
Add sequences together and see new ones appear, like odd numbers adding up to squares.
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Add up odd numbers, and you get squares.
Sometimes, number sequences can be related to each other in surprising ways.
What happens when we start adding up odd numbers? Let's take a look at the sums of consecutive odd numbers starting from 1.
Count the dots row by row to see the square.
Count the number of dots in each row and add them together. This sum results in a square number:
Worked example: count up and down to make a square.
Find the resulting number sequence when you start adding the counting numbers up, and then add them back down. Does this relate to any sequence we already know?
Add two consecutive triangular numbers, step by step.
Let's investigate what happens when we add pairs of consecutive triangular numbers. The first few triangular numbers are 1, 3, 6, 10, and 15.
If we take the first pair, , the sum is .
For the next pair, , the sum is .
Continuing this pattern with , we get 16, and gives us 25. Looking at the resulting sums (4, 9, 16, 25...), we can see a clear and beautiful pattern emerging.
This shows that adding consecutive triangular numbers produces the sequence of numbers.
Find the sum of the first 100 odd numbers.
What is the sum of the first 100 odd numbers ()?