Find a polynomial function f(n) such that f(1), f(2), ... , f(8) is the following sequence.

Question

4, 12, 20, 28, 36, 44, 52, 60

f(n) =___ans________

Answer 1

First we find out what degree the polynomial is:

Since when x = 1, f(x) = 4, and when x = 2, f(x) = 12 the amount of change is 8

And when x = 1, f(x) = 4, and when x = 2, f(x) = 12 the amount of change is 8, the rate of change is constant.

The equation is of the order f(x) = ax + b

so setting up two equations:

a(1) + b = f(x) = 4 and a(2) + b = f(x) = 12

substituting the first equation into the second 

a + b = 4, 2a + b = 12

b = 4 - a

2a + (4 - a) = 12

a = 8

putting this value into the original equation gives

a + b = 4

8 + b = 4

b = -4

so verifying

f(1) = 8(1) - 4, f(1) = 8 - 4 = 4

f(2) = 8(2) - 4, f(2) = 16 - 4 = 12

f(3) = 8(3) - 4, f(3) = 24 - 4 = 20

f(4) = 8(4) - 4, f(4) = 32 - 4 = 28

f(5) = 8(5) - 4, f(5) = 40 - 4 = 36

 

you can finish from here

 

Answer 2

Arithmetic sequence

a_n = a₁ + (n - 1)d

f(n) = 4 + 8(n - 1)