Triangular Numbers
Find at least 5 different ways to develop (and prove) the formula for the sum of the first n natural numbers.
T(n) is the n th Triangular Number. Perhaps they are so named because they can be represented by a triangular array of dots:
Hints:
Try to find approaches that are
Numerical
For example, write the sum "forward" and "backward". Then add the respective sides of the two equations
Geometric
Algebraic
Induction
Uses other known formulas.
Arithmetic Mean
Sum of an Arithmetic Series
NOTE THAT for all n,
That is, the n th square number is the sum of two trianglular numbers, the n th triangular number and the previous triangular number.
