2)

Explanation: When matrix P is powered by 3 it gives a result of , when matrix P is powered by 4 it gives a result of, and when matrix P is powered by 5 it gives a result of . The pattern shown is that results have a common factor of such as 4 shown in matrix, 8 shown in matrix, and then16 shown in matrix. Once the resulting matrix is factor out, the left over numbers inside the matrix has a general pattern of either such as 9 when n = 3, 17 when n = 4 and then 33 when n = 5 or such as 7 when n = 3, 9 when n = 4 and then 31 when n = 5. Thus, these results make a general expression that. This formula is proven correct because when from GDC and this formula also works whenfrom GDC.

Explanation: When matrix S is powered by 3 it gives a result of from GDC, when matrix S is powered by 4 it gives a result offrom GDC, and when matrix S is powered by 5 it gives a result of from GDC. The pattern shown is that results have a common factor of such as 4 shown in matrix, 8 shown in matrix, and then16 shown in matrix. Once the resulting matrix is factor out, the left over numbers inside the matrix has a general pattern of either such as 28 when n = 3, 82 when, and then 244 when n = 5 or such as 26 when n = 3, 80 when n = 4, and then 242 when n = 5. Thus, these results make a general expression that. This formula is proven correct because when from GDC and this formula also works whenfrom GDC.

3) When , , , ,

The pattern shows that the results of is equivalent to result matrix, where as the results of is equivalent to result matrix and the results of is equivalent to result matrix. The pattern also shows that whenthe matrix formula is, when the matrix formula is, and when the matrix formula is. This shows that any number representing corresponds toand. Thus, these results give a general expression that. For instance, if and then = from GDC.

4)

Explanation: After a further investigation with further values of k and n, the results shows that the limitation for n is that all negative numbers and non-integers value because when a matrix is powered by negative numbers or non-integers value the calculator gives a math error text. In addition there’s no limitation for k.

5)

In conclusion,