2 questions I need some help with please!
1. Find the partial sum S5.
an= -2n+16
2.Solve the system by triangularizing the augmented matrix and using bakc substitution.
7x=y-6
3y= -5-2x
Sure! I can help you with both questions. Let's start with the first one.
1. Finding the partial sum S5:
In order to find the partial sum S5, we need to find the sum of the first 5 terms of the sequence. The sequence is defined by the formula an = -2n + 16.
To find the terms of the sequence, substitute the values of n from 1 to 5 into the formula and calculate the corresponding terms:
a1 = -2(1) + 16 = 14
a2 = -2(2) + 16 = 12
a3 = -2(3) + 16 = 10
a4 = -2(4) + 16 = 8
a5 = -2(5) + 16 = 6
Now, let's find the sum of these terms:
S5 = a1 + a2 + a3 + a4 + a5
= 14 + 12 + 10 + 8 + 6
= 50
Therefore, the partial sum S5 for the given sequence is 50.
Now let's move on to the second question.
2. Solving the system by triangularizing the augmented matrix and using back substitution:
The system of equations is:
7x = y - 6
3y = -5 - 2x
To solve this system using triangularization, we need to write the system of equations in augmented matrix form. The augmented matrix for this system is:
[ 0 | 1 | -6 ]
[ -2 | 3 | -5 ]
To triangularize the matrix, we can perform row operations to eliminate coefficients below the diagonal. The goal is to get a matrix in the form:
[ a | b | c ]
[ 0 | d | e ]
Let's perform the necessary row operations:
R2 = (3 * R2) + (2 * R1) (Multiply R2 by 3 and add 2 times R1 to it)
= [ 0 | 1 | -6 ]
[ 0 | 9 | -17 ]
Now, we have a triangular matrix. We can now use back substitution to find the values of x and y.
From the second row, we have:
9y = -17
Solving for y, we get:
y = -17/9
Substituting this value of y into the first row, we have:
0x + 1y = -6
y = -6
Substituting the value of y = -6 back into the first equation, we get:
7x = -6 - 6
7x = -12
Solving for x, we get:
x = -12/7
Therefore, the solution to the system is x = -12/7 and y = -17/9.
I hope this helps! Let me know if you have any further questions.