▁ ▂ ▄ ▅ ▆ ▇ █ тower oғ нanoι reғlecтιon █ ▇ ▆ ▅ ▄ ▂ ▁

Starting off, within the concept of mathematical induction, there are two major steps. The first is "Show true for n=1" and the second is "Assume the statement is true for n=k, and prove true for k+1." 

Using such steps in an example...

If the equation or problem given is 1 + 2 + 3 +4... + (2n-1) = n^2 , one would first begin by proving that n=1. To do this, you would plug in a "1" for anywhere you see a "n" within the equation. Therefore it would look 2(1) - 1, which is equal to 1 - and then (1)^2, which is also equal to one. By doing that, we have proved that both the left hand side and right hand side are equal to 1, showing true that n=1.

After that, we would continue on to step 2, which is "assume the statement is true for n=k, and prove true for k+1." This means that first, we will plug in a "k" for anywhere we see a "n" in the problem. Upon doing so, the resulting equation would be:

 1 + 2 + 3 +4... + (2k-1) = k^2  (This is known as the recursive formula)

After you find the equation above, you then plus in a "k+1" for anywhere a k is present. Therefore, the next resulting equation would be:

 1 + 2 + 3 +4... + [2(k+1)-1] = (k+1)^2

You then simplify the equation and plug in the recursive formula (mentioned above). This indicates that the equation will become:

k^2 + 2k +2-1 = (k+1)^2

and then simplifying it down

k^2 +2k + 1 = (k+1)^2

which is

(k+1)^2 = (k+1)^2

Thus we have shown that k+1 is true. It follows from Mathematical Induction that the statement is true for every positive integer.

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Moving along, in class, we did a project called "The Tower of Hanoi." This particular project was to find the correlation between the number of rings and the amount of times it was moved. Starting with 2, then 3, then 4, and then maxing at 7 rings, with three poles; my group discovered that the equation for the correlation was:

T1 = 2^n - 1

and the recursive of it was:

Tn = 2Tn +1

An observation i concluded while experimenting with the discs is that if you move the lowest number to the second pole, and then move the second lowest number to the third pole, you can then put the one on the second pole on the third pole - therefore, giving room for the third to be moved to the second pole. By using such a strategy, it is easier to move all the discs from the first pole to the third pole.