Answer:
8. Arithmetic Progression
9. [tex]f(9) = 300[/tex]
Step-by-step explanation:
Given
[tex]\{(1,55)\ (2, 45)\ (3, 35)\ (4, 25)\}[/tex]
Solving (8): Arithmetic or Geometric
We start by checking if it is arithmetic by checking for common difference (d).
[tex]d = y_2 - y_1 = y_3 - y_2 = y_4 - y_3[/tex]
This gives:
[tex]d = 45 - 55 = 35 - 45 = 25 - 35[/tex]
[tex]d = -10 = -10 = -10[/tex]
[tex]d=-10[/tex]
Because the common difference is equal, then it is an arithmetic progression
Solving (8):
[tex]f(n) = f(1) + f(n-1)[/tex]
To find f(9), we substitute 9 for n
[tex]f(9) = f(1) + f(9-1)[/tex]
[tex]f(9) = f(1) + f(8)[/tex]
We need to solve for f(8); substitute 8 for n
[tex]f(8) = f(1) + f(8 - 1)[/tex]
[tex]f(8) = f(1) + f(7)[/tex]
We need to solve for f(7); substitute 7 for n
[tex]f(7) = f(1) + f(7 - 1)[/tex]
[tex]f(7) = f(1) + f(6)[/tex]
We need to solve for f(6); substitute 6 for n
[tex]f(6) = f(1) + f(6 - 1)[/tex]
[tex]f(6) = f(1) + f(5)[/tex]
We need to solve for f(5); substitute 6 for n
[tex]f(5) = f(1) + f(5 - 1)[/tex]
[tex]f(5) = f(1) + f(4)[/tex]
From the function, f(4) = 25 and f(1) = 55.
So:
[tex]f(5)=55 + 25[/tex]
[tex]f(5)=80[/tex]
[tex]f(6) = f(1) + f(5)[/tex]
[tex]f(6) = 55 + 80[/tex]
[tex]f(6) = 135[/tex]
[tex]f(7) = f(1) + f(6)[/tex]
[tex]f(7) = 55 + 135[/tex]
[tex]f(7) = 190[/tex]
[tex]f(8) = f(1) + f(7)[/tex]
[tex]f(8) = 55 + 190[/tex]
[tex]f(8) = 245[/tex]
[tex]f(9) = f(1) + f(8)[/tex]
[tex]f(9) = 55 + 245[/tex]
[tex]f(9) = 300[/tex]
