Question

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

Answer 1

$f(1)=3\ and\ f(n)=-2f(n-1)+1\\f(2)=-2f(2-1)+1=-2f(1)+1=-2(3)+1=-6+1=\boxed{-5}\\f(3)=-2f(3-1)+1=-2f(2)+1=-2\boxed{(-5)}+1=10+1=\boxed{11}\\f(4)=-2f(4-1)+1=-2f(3)+1=-2\fbox{11}+1=-22+1=\fbox{-21}\\f(5)=-2f(5-1)+1=-2f(4)+1=-2\fbox{(-21)}+1=42+1=\boxed{\boxed{43}}$

Answer 2

Our function is $f(n)=-2f(n-1)+1$.
We want to find f(5).
Let's substitute 5 for n.
f(5) = -2f(5-1) + 1
f(5) = -2f(4) + 1
We need to know f(4).
f(4) = -2f(4-1) + 1
f(4) = -2f(3) + 1
We need to know f(3).
f(3) = -2f(3-1) + 1
f(3) = -2f(2) + 1
We need to know f(2).
f(2) = -2f(2-1) + 1
f(2) = -2f(1) + 1
We know that f(1) = 3
f(2) = -2(3) + 1
f(2) = -6 + 1
f(2) = -5
Now we can use f(2) = -5 to find f(3)...
f(3) = -2(-5) + 1
f(3) = 10 + 1
f(3) = 11
Now we can use f(3) = 11 to find f(4)...
f(4) = -2(11) + 1
f(4) = -22 + 1
f(4) = -21
Now we can use f(4) = -21 t find f(5)!
f(5) = -2(-21) + 1
f(5) = 42 + 1
f(5) = 43