Students can demonstrate how adding specific sequences generates entirely new sequence types.
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Hook showing how adding odd numbers results in 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.
Diamond dot pattern showing the up and down addition.
Count the number of dots in each row and add them together. This sum results in a square number:
Worked example showing how counting up and down makes squares.
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?
Faded step-by-step for adding consecutive triangular numbers.
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.
Practice calculating the sum of 100 odd numbers.
What is the sum of the first 100 odd numbers ()?