…………………………………………………………………………………………………………………………………………………….
2. Consider the matrices P = and S =
Calculate Pn and Sn for other values of n and describe any pattern(s) you observe.
P²=*== Determinant: 10036=64
P³=*= = Determinant: 1296784=512
P⁴=*== Determinant: 1849614400=4096
P⁵=*= Determinant: 278784246016=32768
S²=*== Determinant: 400256=144
S³=*== Determinant: 1254410816=1728
S⁴=*== Determinant: 430336409600=20736
S⁵=*= =
Determinant: 1524121614992384=248832

To obtain the matrices seen above I used the matrix function on the TI 84 calculator. In order to obtain the discriminate in each problem I used the formula ad –bc such that.
Patterns Found:
The patterns portrayed in matrices Pⁿ and Sⁿ mainly correspond to the coefficient shown in the box and the determinant in each individual series. In the series of Pⁿ, as each exponent progress consecutively the coefficients associated with that particular matrix (in factored form) are multiplied by two (2, 4, 8, and 16). Also, each determinant found in the Pⁿ series is multiplied by 8 as each exponent progress consecutively (64, 512, 4096, 32768). In the Sⁿ each determinant found yielded is multiplied by 12 as each exponent progress consecutively (512, 1728, 20736, 248832.) However, unlike the pattern found in the Pⁿ series, there are no consistencies in regards to coefficients of the matrixes factored form.
………………………………………………………………………………………………………………………………………………......
3. Now consider matrices of the form
Steps 1 and 2 contain examples of these matrices for k = 1, 2 and 3
K=1: Determinant: 40=4 _ K=2: Determinant: 91=8
K=3: Determinant: 164=12
Consider other values of k, and describe any pattern(s) you observe.
K=4: = Determinant: 259=16
K=5: = Determinant= 3616=20
K=6: = Determinant= 4925=24
K=7: = Determinant= 6432=28
Patterns found:
The pattern that was evident in the above matrices corresponds to the determinant value as well as the value of the consecutive matrices. As each value for K is raised consecutively the numbers in the associated matrix is then raised by one. The determinant also followed a pattern by raising the K value consecutively the determinant also rose by four.
Generalize these results in terms of k and n.
n=exponent value
Results for k, n=2: =*==
Results for k, n=3: = *=
Results for k, n=4 = *=
Results for k, n=5 = *=
 Therefore, based on the matrices shown above, the generalized formula of this pattern is
4. Use technology to investigate what happens with further values of k and n. State the scope or limitations of k and n.
Results for k=2, n=(2) =
Results for k=2, n=(3) =
Results for k=(2), n=(2) =
Results for k=(3), n=(2) =
Results for k=(.5), n=(2) =
Results for k=(2), n=(.5) =
 According to the numbers substituted in the problem above, it can be inferred that the scope of the problem is all real numbers and fractions. I have found no limitations that hinder the use of k and n.
5. Explain why your results hold true in general.
Referring back to work shown in question three, my results for these statements hold true because of the amount of times the equations were carried out. Overall, the steadiness of the results for each of the problems ensures the validity of each statement. A wide array of aspects was used to prove these statements true ranging from negative integers to fractions. The patterns seen were also consistent with the results which reinforce the strength of my statements. Therefore, I conclude that my results hold true in general.