if f(1) = 3 and f(n) = -2f(n-1) + 1, then f(5) =

Question

help show all neccessary steps please

Answer 1

Hello Leon,

This is a recursive formula which means it requires a previous output of the function to determine the next output. So find f(2) based on the fact that you know what f(1) is. Then you can find f(3), then f(4), and finally f(5).

I am available if you would like any one-on-one time. Don't be afraid to ask!

Answer 2

METHOD 1:

We start with f(n) = - 2f(n-1) + 1 and plug in n = 2 which yields

f(2) = - 2f(1) + 1

It is given that f(1) = 3. Therefore, we have

f(2) = - 2(3) + 1 = - 6 + 1 = -5

Now that we have f(2), we can plug in n = 3 in the given equation:

f(3) = - 2f(2) + 1 = - 2(-5) + 1 = 10 + 1 = 11

Going on, we plug in n = 4 in the given equation:

f(4) = - 2f(3) + 1 = - 2(11) + 1 = - 22 + 1 = -21

Finally, we plug in n = 5 in the given equation:

f(5) = - 2f(4) + 1 = - 2(-21) + 1 = 42 + 1 = 43

So, f(5) = 43

METHOD 2:

Plug in n = 5 in the given equation to get

f(5) = -2f(4) + 1

f(5) = -2(-2f(3) + 1) + 1

f(5) = 4f(3) - 1

f(5) = 4(-2f(2) + 1) - 1

f(5) = -8f(2) + 3

f(5) = -8(-2f(1) + 1) + 3

f(5) = 16f(1) - 5

f(5) = 16(3) - 5

f(5) = 48 - 5

f(5) = 43

METHOD 3: (Preferred)

Inspection yields that f(n) obeys the following formula:

f(n) = (-2)^n-1 f(1) + (1 - (-2)^n-1)/3

Plug in f(1) = 3 and n = 5

f(5) = (-2)⁴ (3) + (1 - (-2)⁴)/3 = (16)(3) + (1 - 16)/3 = 48 - 5 = 43