Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
When you add the Counting numbers up, you get the Triangular numbers sequence.
Cumulative sums: 1 = 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10 ...
Triangular numbers sequence: 1, 3, 6, 10, 15, ...
Pictorial explanation: Each triangular number can be shown as dots arranged in a triangle.
First triangular number: 1 dot
Second triangular number: 1 + 2 = 3 dots forming a small triangle
Third triangular number: 1 + 2 + 3 = 6 dots forming a larger triangle
Each new row adds the next counting number of dots, building a triangular shape.
Explanation
The chapter introduces triangular numbers as 1, 3, 6, 10, 15, 21, 28, ... These are exactly the cumulative sums of counting numbers: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, and so on. The pictorial representation shows dots arranged in triangular patterns where each row has one more dot than the row above it.
Solution Steps
Add counting numbers cumulatively: 1, 1+2, 1+2+3, 1+2+3+4, ...
The results form the Triangular numbers sequence: 1, 3, 6, 10, 15, ...
Each triangular number can be represented pictorially as dots arranged in a triangle.
Each new row adds the next counting number of dots.