, where Un=a specific term
a=first term
r=common ratio (multiplier between the entries of the geometric sequence)
n=the number of the specific term (with relation to the rest of the sequence).
For Yn:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, 2, 4, 8, … These terms represent the pattern between the scalar values multiplied to
Y= to achieve an end product of Yn.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 2, 4, 8}, a=1
r=2
However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, Y. In order to find the final expression for Yn, we must multiply the general scalar value 2n-1 by matrix Y:
In order to test the validity of this expression, we can employ it to find Y5. Using the previous method of calculations, we find that Y5=. (We can also check this on a calculator).
Now, using the expression, we find the value of Y5 to be the same (proving the accuracy of our expression):
Y5=25-1
=24
=16
=
We conclude that is indeed a valid expression.
To find an expression for (X+Y)n, we must first determine the patterns by calculating the values of (X+Y), (X+Y)2, (X+Y)3, (X+Y)4.
For (X+Y)n, where n=1, i.e. (X+Y):
(X+Y)=
=
=
=2I
Where n=2, i.e. (X+Y)2:
(X+Y)2=(X+Y)(X+Y)
(X+Y)2=
=
=
=
=4I
Where n=3, i.e. (X+Y)3:
(X+Y)3=(X+Y)(X+Y)(X+Y)
(X+Y)3=(X+Y)2(X+Y)
(X+Y)3=
=
=
=
=8I
Where n=4, i.e. (X+Y)4:
(X+Y)4=(X+Y)(X+Y)(X+Y)(X+Y)
(X+Y)4=(X+Y)3(X+Y)
(X+Y)4=
=
=
=
=16I
We can now find an expression for (X+Y)n through consideration of the integer powers of (X+Y) and using the general equation for the geometric sequence.
, where Un=a specific term
a=first term
r=common ratio (multiplier between the entries of the geometric sequence)
n=the number of the specific term (with relation to the rest of the sequence).
For (X+Y)n:
When n=1, 2, 3, 4, … (integer powers increase), then:
1, 2, 4, 8, … These terms represent the the scalar values multiplied to the matrix
(X+Y)= to achieve an end product of (X+Y)n.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 2, 4, 8}, a=1
r=2
However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, (X+Y) in order to yield the product (X+Y)n. In order to find the final expression for (X+Y)n, we must multiply the general scalar value 2n-1 by matrix (X+Y):
, which can also be written as .
In order to test the validity of this expression, we can employ it to find (X+Y)5. Using the previous method of calculations, we find that (X+Y)5=. (We can also check this on a calculator).
Now, using the expression, we find the value of (X+Y)5 to be the same (proving the accuracy of our expression):
(X+Y)5=25-1
=24
=16
=
We conclude that is indeed a valid expression.
Let A=aX, where a is a constant. Using different values of a, we can calculate A2, A3, A4:
In order to ensure the validity of our findings, we will use 3 different values of a (2, -2, and 10).
When a=2, i.e. A=2X:
A=2
A=
When a=2, i.e. A2=(2X)2:
A2=A●A
=
=
=
When a=2, i.e. A3=(2X)3:
A3=A2●A
=
=
=
When a=2, i.e. A4=(2X)4:
A4=A3●A
=
=
=
When a=-2, i.e. A= -2X:
A= -2
A=
When a= -2, i.e. A2=(-2X)2:
A2=A●A
=
=
=
When a= -2, i.e. A3=(-2X)3:
A3=A2●A
=
=
=
When a= -2, i.e. A4=(-2X)4:
A4=A3●A
=
=
=
When a=10, i.e. A=10X:
A=10
A=
When a=10, i.e. A2=(10X)2:
A2=A●A
=
=
=
When a=10, i.e. A3=(10X)3:
A3=A2●A
=
=
=
When a=10, i.e. A4=(10X)4:
A4=A3●A
=
=
=
Through consideration of the integer powers of A (2, -2, and 10), we can observe a pattern and find an expression for An.
For An when a=2:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, 4, 16, 64, … These terms represent the pattern between the scalar values multiplied to
A=aX where a=2 and hence A= to achieve an end product of An.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 4, 16, 64}, f=1
r=4
Here, we can express r in terms of a (a=2):
r=4
4=2●2
r=2a
We can also express in terms of a:
For An when a= -2:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, -4, 16, -64, … These terms represent the pattern between the scalar values multiplied to
A=aX where a= -2 and hence A= to achieve an end product of An.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, -4, 16, -64}, f=1
r= -4
Here, we can express r in terms of a (a= -2):
r= -4
-4= -2●2
r=2a
We can also express in terms of a (a= -2):
For An when a=10:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, 20, 400, 8000, … These terms represent the pattern between the scalar values multiplied to
A=aX where a=10 and hence A= to achieve an end product of An.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 20, 400, 8000}, f=1
r=20
Here, we can express r in terms of a (a=10):
r= 20
20= 10●2
r=2a
We can also express in terms of a (a=10):
Using all three values of constant a (2, -2, and 10), we have arrived at a general equation. However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, A. In order to find the final expression for An, we must multiply the general scalar value by A=aX:
, which can also be expressed as .
Let B=bY, where b is a constant. Using different values of b, we can calculate B2, B3, B4:
In order to ensure the validity of our findings, we will use 3 different values of b (2, -2, and 10).
When b=2, i.e. B=2Y:
B=2
B=
When b=2, i.e. B2=(2Y)2:
B2=B●B
=
=
=
When b=2, i.e. B3=(2Y)3:
B3=B2●B
=
=
=
When b=2, i.e. A4=(2Y)4:
B4=B3●B
=
=
=
When b=-2, i.e. B= -2Y:
B= -2
B=
When b= -2, i.e. B2=(-2Y)2:
B2=B●B
=
=
=
When b= -2, i.e. B3=(-2Y)3:
B3=B2●B
=
=
=
When b= -2, i.e. B4=(-2Y)4:
B4=B3●B
=
=
=
When b=10, i.e. B=10Y:
B=10
B=
When b=10, i.e. B2=(10Y)2:
B2=B●B
=
=
=
When b=10, i.e. B3=(10Y)3:
B3=B2●B
=
=
=
When b=10, i.e. B4=(10Y)4:
B4=B3●B
=
=
=
Through consideration of the integer powers of B (2, -2, and 10), we can observe a pattern and find an expression for Bn.
For Bn when b=2:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, 4, 16, 64, … These terms represent the pattern between the scalar values multiplied to
B=bY where b=2 and hence B= to achieve an end product of Bn.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 4, 16, 64}, f=1
r=4
Here, we can express r in terms of b (b=2):
r=4
4=2●2
r=2b
We can also express in terms of b:
For Bn when b= -2:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, -4, 16, -64, … These terms represent the pattern between the scalar values multiplied to
B=bY where b= -2 and hence B= to achieve an end product of Bn.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, -4, 16, -64}, f=1
r= -4
Here, we can express r in terms of b (b= -2):
r= -4
-4= -2●2
r=2b
We can also express in terms of b (b= -2):
For Bn when b=10:
When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:
1, 20, 400, 8000, … These terms represent the pattern between the scalar values multiplied to
B=bX where b=10 and hence B= to achieve an end product of Bn.
Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:
In the sequence {1, 20, 400, 8000}, f=1
r=20
Here, we can express r in terms of b (b=10):
r= 20
20= 10●2
r=2b
We can also express in terms of b (b=10):
Using all three values of constant b (2, -2, and 10), we have arrived at a general equation. However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, B. In order to find the final expression for Bn, we must multiply the general scalar value by B=bY:
, which can also be expressed as .
To find an expression for (A+B)n, we must first determine the patterns by calculating the values of (A+B), (A+B)2, (A+B)3, (A+B)4.
For (A+B)n, where n=1, i.e. (A+B):
(A+B)1=A+B
Where n=2, i.e. (A+B)2:
(A+B)2=(A+B)(A+B)
=
=
=
=
=
=
=
Where n=3, i.e. (A+B)3:
(A+B)3=(A+B)(A+B)(A+B)
(A+B)3=(A+B)2(A+B)
(A+B)3=
=
= (A+B)
=
But AB and BA both equal 0, as:
A=aX
=a
=
B=bY
=b
=
So AB=
=
=
=0
BA=
=
=
=0
We re-write as
=
=
Where n=4, i.e. (A+B)4:
(A+B)4=( A+B)( A+B)( A+B)( A+B)
(A+B)4=( A+B)3(A+B)
(A+B) 4=(A+B)
=
=
=
=A4+B4
We can now find an expression for (A+B)n through consideration of the integer powers of (A+B) and observing the pattern that has emerged.
(A+B)=A+B
(A+B)2=A2+B2
(A+B)3=A3+B3
(A+B)4=A4+B4, and so on.
Thus, we can conclude that (A+B)n=An+Bn, which can be rewritten as follows (to be expressed in terms of X and Y):
(A+B)n=An+Bn
=(aX)n+(bY)n
=anXn+bn+Yn
Now we consider M=, where M=A+B:
M=A+B
= aX+bY
=+
=
We can also prove that M2=A2+B2 (which we can use to later calculate M3=A3+B3, M4=A4+B4 to find a general statement for Mn, in terms of aX and bY):
M2=A2+B2
=(aX)2+(bY)2
=+
=+
=+
=
Now, tracing the pattern in order to find the expression for Mn, we see that:
M=A+B=
M2=A2+B2=
M3=A3+B3=
M4=A4+B4=
Expressing Mn in terms of aX and bY (=A and B respectively), we have arrived at the general statement:
Mn=(aX)n + (bY)n, which we can also express as
Mn=anXn + bnYn.
