Given f(11)=11 and f(x+3) = (f(x)-1)/(f(x)+1) for all x, find f(2000)
Please don't do this. I will have to destroy your grade kiddo.
Turfe
Hey, thank me in class for doing your work for you :P
Look for the pattern
f(11) = 11
f(14) = (11 - 1)/(11 + 1) = 10/12 = 5/6
f(17) = (5/6 - 1)/(5/6 + 1) = (-1/6) / (11/6) = -1/11
f(20) = (-1/11 - 1)/(-1/11 + 1) = -12/11 / (10/11) = -12/10 = -6/5
f(23) = (-6/5 - 1)/(-6/5 + 1) = -11/5 / (-1/5) = 11
so it repeats every 12 numbers (23 - 11 = 12)
and since 2000 = 165 • 12 + 20, f(2000) will be the same as f(20) = -6/5
:P
To find the value of f(2000), we need to use the given equation f(x+3) = (f(x)-1)/(f(x)+1) and make a sequence of substitutions until we arrive at f(2000).
Let's start by using the given value f(11) = 11. We can substitute x = 11 into the equation:
f(11+3) = (f(11)-1)/(f(11)+1)
Simplifying this equation gives us:
f(14) = (11-1)/(11+1)
f(14) = 10/12
f(14) = 5/6
Now, we can use f(14) to find f(17). Substitute x = 14 into the equation:
f(14+3) = (f(14)-1)/(f(14)+1)
Simplifying this equation gives us:
f(17) = (5/6-1)/(5/6+1)
f(17) = (-1/6)/(11/6)
f(17) = -1/11
Next, we can use f(17) to find f(20). Substitute x = 17 into the equation:
f(17+3) = (f(17)-1)/(f(17)+1)
Simplifying this equation gives us:
f(20) = (-1/11-1)/(-1/11+1)
f(20) = (-12/11)/(-10/11)
f(20) = 12/10
f(20) = 6/5
We continue this process, substituting the previous value into the equation, until we reach f(2000).
f(23) = (f(20)-1)/(f(20)+1) = (6/5-1)/(6/5+1) = (1/5)/(11/5) = 1/11
f(26) = (f(23)-1)/(f(23)+1) = (1/11-1)/(1/11+1) = (-10/11)/(12/11) = -10/12 = -5/6
f(29) = (f(26)-1)/(f(26)+1) = (-5/6-1)/(-5/6+1) = (-11/6)/(-1/6) = 11/1 = 11
The pattern repeats as we shift x by 3, and we keep alternating between positive and negative values. Using this pattern, we can determine f(2000).
f(29+3) = (f(29)-1)/(f(29)+1)
f(32) = (11-1)/(11+1)
f(32) = 10/12
f(32) = 5/6
Now, since f(32) = 5/6, we can determine f(35).
f(35) = (5/6-1)/(5/6+1)
f(35) = (-1/6)/(11/6)
f(35) = -1/11
From here, we observe that the pattern repeats. Since f(35) = -1/11, we can conclude that f(38) = -10/11, f(41) = 6/5, and so on.
To find f(2000), we need to determine the position of 2000 in the sequence. The position can be calculated using the formula:
position = (x - 11)/3
In this case, x = 2000, so the position is:
position = (2000 - 11)/3
position = 1989/3
position ≈ 663
Since we know that the pattern repeats every three steps, we divide the position by 3 to determine the number of cycles:
cycles = position/3
cycles ≈ 663/3
cycles ≈ 221
We have completed 221 cycles, which brings us to the position f(11+3*221) = f(674), where f(11) = 11.
Therefore, f(2000) = f(674) = 11.
Hence, the value of f(2000) is 11.