Asked by Francesca
                Find f(1), f(2), and f(3) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . .
• f(n+1) = f(n) + 2
So, would it be f(n) = f(n+1) + 2? Or would I just keep it like the original and plug in 1, 2, 3. Thanks for any helpful replies.
            
            
        • f(n+1) = f(n) + 2
So, would it be f(n) = f(n+1) + 2? Or would I just keep it like the original and plug in 1, 2, 3. Thanks for any helpful replies.
Answers
                    Answered by
            MathMate
            
    As in algebra, put your known values on the right hand side, and the quantity to be evaluated on the left.  Follow the definition of the function.
So if f(0)=1, then n=0, therefore
f(1)=f(0)+2
...
    
So if f(0)=1, then n=0, therefore
f(1)=f(0)+2
...
                    Answered by
            Damon
            
    just add 2 each time
f(0) = 1
f(1) = 3
f(2) = 5
f(3) = 7 etc
That is an arithmetic sequence
f(n) = 1 + 2n
    
f(0) = 1
f(1) = 3
f(2) = 5
f(3) = 7 etc
That is an arithmetic sequence
f(n) = 1 + 2n
                    Answered by
            Francesca
            
    Ok thank for the responses, but there seems to be a contradiction between the two. Wouldn't f(1) = 1 + 2, which equals 3? 
    
                    Answered by
            Francesca
            
    The f(n+1) is throwing me off what does that mean?
    
                    Answered by
            MathMate
            
    As Damon mentioned, it is an arithmetical sequence (or in general, any sequence).
So the zeroth term is denoted f(0), the 1st term f(1), etc.
So for
1,3,5,7,9,...
f(0)=1
f(1)=3
f(2)=5
...
the relation between them is therefore
f(n+1)=f(n)+2
This is called a recurrence relation, and you can only evaluate a certain term if the previous terms are know.
For the Fibonacci sequence, it would be
f(n)=f(n-1)+f(n-2)
f(0)=0
f(1)=1
so
f(2)=f(1)+f(0)=1+0=1
f(3)=f(2)+f(1)=1+1=2
f(4)=f(3)+f(2)=2+1=3
f(5)=f(4)+f(3)=3+2=5
....
to give the Fibonacci sequence
1,1,2,3,5,8,13,21,34....
    
So the zeroth term is denoted f(0), the 1st term f(1), etc.
So for
1,3,5,7,9,...
f(0)=1
f(1)=3
f(2)=5
...
the relation between them is therefore
f(n+1)=f(n)+2
This is called a recurrence relation, and you can only evaluate a certain term if the previous terms are know.
For the Fibonacci sequence, it would be
f(n)=f(n-1)+f(n-2)
f(0)=0
f(1)=1
so
f(2)=f(1)+f(0)=1+0=1
f(3)=f(2)+f(1)=1+1=2
f(4)=f(3)+f(2)=2+1=3
f(5)=f(4)+f(3)=3+2=5
....
to give the Fibonacci sequence
1,1,2,3,5,8,13,21,34....
                    Answered by
            Francesca
            
    OK example: f(n+1) = 3f(n) 
f(1) = 3
f(2) = 6
f(3) = 9
Right?
    
f(1) = 3
f(2) = 6
f(3) = 9
Right?
                    Answered by
            MathMate
            
    Almost! 
What you have done was for
f(n+1)=f(n)+3, and f(1)=3.
so f(2)=f(1)+3=3+3=6
f(3)=f(2)+3=6+3=9.
If you are solving f(n+1)=3f(n) with f(1)=3, then
f(2)=3*f(1)=3*3=9
f(3)=3*f(2)=3*9=27
f(4)=3*f(3)=3*27=81,
...
effectively a geometric sequence instead.
    
What you have done was for
f(n+1)=f(n)+3, and f(1)=3.
so f(2)=f(1)+3=3+3=6
f(3)=f(2)+3=6+3=9.
If you are solving f(n+1)=3f(n) with f(1)=3, then
f(2)=3*f(1)=3*3=9
f(3)=3*f(2)=3*9=27
f(4)=3*f(3)=3*27=81,
...
effectively a geometric sequence instead.
                    Answered by
            Francesca
            
    Oops. . .Sorry disregard previous post. . .
    
                    Answered by
            Francesca
            
    f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . .
 
Find f(1), f(2), f(3)
    
Find f(1), f(2), f(3)
                    Answered by
            MathMate
            
    Is this the previous problem where f(n)=f(n-1)+3, or is this a new problem?
I don't see the recursive definition of f(n).
    
I don't see the recursive definition of f(n).
                    Answered by
            Francesca
            
    Yes, they both follow the same recursive definition. I was just trying the second part on my own to see if I understand. Sorry about the misunderstanding. . .
    
                                                    There are no AI answers yet. The ability to request AI answers is coming soon!
                                            
                Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.