Trapezoidal Numbers – 1
Trapezoidal Numbers are the sum of two or more consecutive numbers. They deserve their name because you
can make a trapezoid with that many dots, as pictured in the examples below. Note that having 1 dot on the top
row is stretching the idea of being a trapezoid a bit, but it is allowed for these numbers.
THE CHALLENGE
Show that every odd number is a Trapezoidal Number.
EXPLORATION
Can you find some even numbers that are Trapezoidal Numbers? Can you find some that aren’t?
Notes
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THE CHALLENGE
Every odd number is of the form 2n + 1. Because 2n + 1 = n + (n + 1), that shows every odd number, larger than 1, is a Trapezoidal number..
EXPLORATION
Among the first few even numbers, we have the following:
- 2: Not trapezoidal
- 4: Not trapezoidal
- 6: 6 = 1 + 2 + 3
- 8: Not trapezoidal
- 10. 10 = 1 + 2 + 3 + 4
- 12: 12 = 3 + 4 + 5
- 14: 14 = 2 + 3 + 4 + 5
- 16: Not trapezoidal
It looks like powers of 2 are never trapezoidal. Will follow that up in a later Puzzle of the Week.