Find at least 5 different ways to develop a closed form formula for the sum of the first

nnatural numbers.Which lead to understanding of the formula?

Which lead to proof of the formula?

## T(n) = 1 + 2 + 3 + 4 + . . . + n

T(n)is thenthTriangular Number. Perhaps they are so named because they can be represented by a triangular array of dots:

or by a triangular array of blocks:

Using blocks to construct representations of triangular numbers can be adapted to using classroom cubes to show these patterns. Or they can be nicely constructed using grid paper.

Hints:Try to find approaches that are

Numerical1. Look for a pattern. For example:

This is an iterative form for generating triangular numbers; we want to find a closed form.

2. For example, write the sum "forward" and "backward". Then add the respective sides of the two equations

3. Implement a spreadsheet to write a column of trianglar numbers

Click

HEREfor an Excel file that shows a graph for the first 40 triangular numbers.

Geometric1. Use the dot patterns above. Take any

twoadjacent triangle numbers and combine the two triangular arrays into a square array.

2. Use the block pattern for this same strategy.

3. Use the dot patterns above. Take ONE tranglular number and double it.

4. Use the block pattern with this same strategy.

Use the Arithmetic Mean

T(n) = 1 + 2 + 3 + . . . + n-2 + n-1 + n

Arithmetic Series

1 + 2 + 3 + 4 + . . . + n is an arithmetic series if n terms with initial term 1 and common difference d = 1.

Algebraic

1.

This approach was presented by Polya in Mathematical Discovery. It has the additional value in that it provide a technique that can be generalized easily for explore closed formulas for the sum of the first n integer squares, the first n integer cubes, etc.

2.

See

Sums of Powers of Integers -- Derivations from Summationsfor more discussion and extension of this strategy to finding the closed form for sums of powers of integers.

Mathematical Induction

Mathematical induction provides one means for proof of the closed form formula. It does not provide a means to

derivethe formula, but rather to confirm via proof.Use mathematical induction to prove the formula for T(n).

Finite Difference Methods

See Using the Technique of Finite Differences: Examples and Problems

The case for T(n) is one of the examples.

Return to the