Understanding Hexagonal Numbers
Hexagonal numbers represent a figurate number that can form a hexagon. The n-th hexagonal number is given by the formula:
- H(n) = n(2n - 1)
For instance, the first few hexagonal numbers are:
- H(1) = 1
- H(2) = 6
- H(3) = 15
- H(4) = 28
Sequence of Hexagonal Numbers
When you start adding hexagonal numbers, you create a sequence of cumulative sums. Here is how it works:
- S(1) = H(1) = 1
- S(2) = H(1) + H(2) = 1 + 6 = 7
- S(3) = H(1) + H(2) + H(3) = 1 + 6 + 15 = 22
- S(4) = H(1) + H(2) + H(3) + H(4) = 1 + 6 + 15 + 28 = 50
Resulting Sequence
The resulting sequence from adding the first n hexagonal numbers is:
- S(n) = {1, 7, 22, 50, ...}
Pattern and Formula
The pattern observed can be generalized. The sum of the first n hexagonal numbers can be expressed as:
- S(n) = n(2n^2 - n + 1)/3
This represents the cumulative addition of hexagonal numbers, leading to a new sequence.
Conclusion
Adding hexagonal numbers generates a unique sequence that reflects the growing complexity of figurate numbers. This exploration connects number theory with geometric shapes, showcasing the beauty of mathematics.