What is the value of f(5) if f(1)=3 and f(x)=3-2f(x-1)?

Question

What is the value of f(5) if f(1)=3 and f(x)=3-2f(x-1)?

Please show the steps for this recursive function. I appreciate it.

Answer 1

They kind of needlessly complicated it.

Let's define a new variable, u=x-1, or x=u+1. Putting this in to the recursive definition, we get

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

Note that the function itself is still the same, it's only the way we give it an input is different. If u=1, it's still true that f(1)=3; it matters not that we relabeled the input to u.

So, we've got f(1), we can get f(2) using

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

And then we just keep churning

f(3)=f(2+1)=3-2f(2)=3-2(-3)=3+6=9

f(4)=f(3+1)=3-2f(3)=3-2(9)=3-18=-15

f(5)=f(4+1)=3-2f(4)=3-2(-15)=3+30=33

Answer 2

Using a RECURSIVE definition of a function, we may start with f(1) [or a₁], and find values up to f(2) [or a₂].

However, most real problems use recursion the other way -- from f(5) to f(1) [or a₅ to a₁].

So, if the recursive definition is: f(x) = 3 - 2f(x-1) ... and f(1) = 3 , here's how we find f(5):

f(5) = 3 - 2 f(4) [note: x=5]

f(5) = 3 - 2 *( 3 - 2 f(3) ) [use constant 5 on left side and use x-4 on right side]

f(5) = 3 - 2 *( 3 - 2*( 3 - 2f(2) ) ) [replace f(3)]

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

f(5) = 3 - 2 *( 3 - 2*( 3 - 2*(3 -2*3) ) )

f(5) = 3 - 2 *( 3 - 2*( 3 - 2*(3 - 6) ) )

f(5) = 3 - 2 *( 3 - 2*( 3 - 2(-3) ) )

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

f(5) = 3 - 2 *( 3 - 2*(9) )

f(5) = 3 - 2 *(-15)

f(5) = 3 + 30

f(5) = 33

One of the easy applications of RECURSION is to define the sum of the numbers from 1 to n as n + the sum of the numbers from 1 to (n-1):

S₁ = 1 ... the sum fo the numbers from 1 to 1 is 1

S_n = n + S(n-1)

So, the sum of the numbers from 1 to 5 is 5 plus the sum of the numbers from 1 to 4. ... keep going until you get to1, then insert S₁= 1 and add up the results.

Answer 3

a_n = 3 - 2(a_n-1)

a₁ = 3

a₂ = 3 - 2(3), a₂ = -3

a₃ = 3 - 2(-3), a₂ = 12

Continue for a₄ and a₅