Today I have for you guys a question from the 2003 Australian Mathematics
Competition and only 3% of students got this one right. So the
question is: if f(11) = 11 and f(n+3) equals f(n)-1 all over f(n)+1
we have to find f(2003). So you might like to pause the video here
and see if you can figure that out. So with a lot of maths questions it helps
to just sit and think for a while and try to come up with a strategy for
approaching it. I don't think this is one of those questions. I think with this one
you just have to charge in and see what happens.
So what I would do is just try out using this formula with n = 11 because we
know what f(11) is. So if we plug in 11 in there we will get f(14). So f(11)
is 11, so just applying that formula we get 10 over 12 and you can simplify that
and turn it into 5 over 6. Okay, maybe we can try doing the next one. So we could
do f(17)... would be the next one if you plug in n = 14. So you've got f(14)
there which is the 5 over 6 that we just found. Now to deal with that kind of
expression where you've got fractions within a fraction, what you want to do is
just multiply top and bottom by 6, and that's gonna get rid of these fractions
within the fraction. So that'll give you 5 - 6 over 5 + 6, so negative 1
over 11. Okay we've got f(17). Still no idea of a strategy for how to work out
f(2003) but let's just keep going and see what happens. So the next one
would be f(20). We plug in our -1/11. That minus one over
-1/11 + 1. Again to get rid of those fractions you want to
multiply top and bottom by 11 which is going to give you -1 - 11
over -1 + 11 which is -12/10 and we can simplify that.
That makes -6/5. Okay that's kind of interesting because
that's the negative reciprocal of that one, and also that's the negative
reciprocal of the original f(11)... was 11. So let's see what happens with the
next number. So we can go up to f(23) which will be the last one minus 1 over that plus
1. Again to get rid of the fractions within the fraction, we multiply top and
bottom by 5 which is going to give you -6 - 5 over -6 + 5 and
then that makes -11/-1 which is just positive 11. So
we've come back to what f(11) was; we started out with 11. And that will mean
that the next one in the sequence, the f(26), well that's just, you know, 11
minus 1 over 11 plus 1 again, just like we had for f(14), so that's gonna be 5/6
and then the next one's going to be -1/11 and then the next
one's gonna be -6/5 and so on. It's gonna just repeat itself. So what
I might do is just write these out in a table. So we had f(11), f(14), f(17),
and f(20), and they were 11... the f(14) was 5/6, then we had -1
over 11, and -6/5, and that's just gonna keep repeating, so
this is also f(23), f(26), f(29), f(32) and so on. So we have to work out f(2003), so
here's where you might want to stop and brainstorm strategies for how to do that.
So the way that I would tackle it is I would try to work out how many threes
have been added to 11 to make 2003, as in this is one 3 added, this is two 3's
added to 11, this is three 3's added, and so on. And then the f(23) which is
four 3's added was 11 again, and then you'll have five 3's added, six, seven, eight, and
so on, and it keeps wrapping around like that. So you'll notice that like all the
multiples of 4 give you 11. All the ones that are a multiple of 4 plus 1, they
give you 5/6 and so on. So if we can work out the number of 3's added
first of all, so let's do that. The way that I would work that out is I would do
2003 minus 11, that gives 1992, and then divide that by 3, so 3 into 1992... not
allowed calculators for this competition, so...
Okay so 664. Now we got to work out where that number goes, like is it a multiple
of 4, in which case it would go in this column; is it a multiple of 4 plus
1, in which case it will be in this column and it'll be 5/6, and so
on. So to work out whether that's a multiple of 4 or not you can just
look at the last two digits of a number, because 100 is divisible by 4 so we
can kind of ignore that bit, so 64 is actually a multiple of 4; it's 16 times 4.
So that means it's going to be in this column with all the other multiples of 4,
which means that the answer is going to be 11. Now just to test that out, what I
would do is kind of like repeat this process with another number just to
check, because it's easy to sort of get off by one, so I might pick like the f(26) -
we already know what that is, but I can do exactly the same process that I
did here just to check that that process is correct. So first thing I did was 2003
minus 11 so I'll do 26 minus 11; that makes 15. And then we did that divided by
3, so 15 divided by 3 is 5, and then what I did with that number was I worked
out was it a multiple of 4 or does it have some remainder. So with this 5
that's one 4 with a remainder of 1, so that would put it in that column which is
correct. So I can see that that process is correct; I'm not off by one. So I'm
pretty confident that 11 is the answer from that. If you have any questions then
leave me a comment below. And give this video a like if you want me to make more
like this. Next time in this series I'll be tackling this question so stay tuned for
that one.