Ideas in Geometry/Instructive examples/Section 2.4:

In the problem Section 2.4: #24, the goal is to determine how the picture shown below "proves" that: 1 + 3 + 5 + ... + (2n-1) = n^2.

The picture below is of my own creation and I uploaded it on my own:

In the problem, we see that each level of the dots on the picture increases by 2 at every level (each level is indicated by the lines dividing the respective sections). If the goal were to find the total number of dots, it could be done by adding up the respective sections, i.e. 1 + 3 + 5, which would reveal the total number of dots between the first 3 levels to be 9.

If it is found that the first three levels would be viewed as n, it is possible that this amount of dots could also be found by using n-squared. This is found to be true as the third level works for this respective formula: 3-squared = 9, which equals the number of dots at the 3rd level and those that have come before it. This also holds true for the other levels: take 6 for example, where n = 6 and therefore the total number of dots found in levels 1-6 would total up to 36, or 6^2. This formula continues to work for each respective level: 4th- 4-squared = 16 dots, 5th- 5-squared = 25 dots, 8th- 8-squared = 64 dots, which equals the total number of dots in the picture below.

Another way of going about this would be to find a formula to find the dots at each individual level, which leads to the formula 2n - 1. By using this formula, with n, the respective level, the number of dots for that respective level can be found, which is shown in the picture below.

There are numerous ways to go about finding out how this picture proves the respective equation and each of the above solutions will successfully do so. It becomes your choice of which one fits your fancy for this particular problem.

Created by TMcCormick31 06:37, 25 October 2010 (UTC)