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Combinatorics

I’m quite unfamiliar with these types of questions, my typical approach would be trial and error to find an pattern to derive a sequence which tends to waste time. In the question attached I don’t understand the idea of introducing a new (m) to find ln in terms of n

(edited 2 years ago)

Reply 1

mqb2766

It just makes it easier to compute the (slightly modified) geometric sequence which represents the number of commands. You have
n: 0, 1, 2, 3, 4
l: 1, 4, 10, 22, 46, ...
So the difference is
3, 6, 12, 24
which is clearly a 3*2^n geometric sequence

They told you to consider the sequence
m: 3, 6, 12, 24,
rather than l. Clearly this is
3*2^n
hence l is
3*2^n - 2

l is a geometric sequence 3*2^n with an additive constant (-2). The difference in successive terms in l shows this as the constant offset disappears due to the subtraction and you're left with a pure geometric sequence. This is what they're hinting at in the definition of m.

(edited 2 years ago)

Reply 2

Ghost of math

Original post by mqb2766

I see now this is similar to the idea of sequences learnt in gcse maths, however how come when I apply this method onto a sequence like the one i wrote,it doesn’t seem to work even though it can be summed geometrically?

Reply 3

mqb2766

Original post by The A.G

In the original mat? question, the offset in the recurrence relationship is constant (+2) where as here it is decaying term (0.5^n) so it is a bit different. Its worth noting that this recurrence relationship represents a simple sum to n (or integral), as there is no multiplier on the a_n term so it just provides a memory on what has already been accumulated. So you're simply summing a decaying exponential.

For this one, if you take differences its 1/2, 1/4, 1/8, ... so it is another representation for a geometric sequence and the closed form is clearly
an = 2 - (1/2)^n
Note the difference doesn't tell you exactly what the geometric sequence is, it tells you that there is an underlying geometric sequence, with possibly an additive offset.

Without going too far into this, the question gives a clear hint about the transformation which gives the standard geometric sequence. You're not necessarily expected to come up with it from scratch, at least in this question.

(edited 2 years ago)

Reply 4

Ghost of math

Original post by mqb2766

Thanks for the explanation, another question I have is for part five how would you go about drawing it. I’ve drawn by literally having to follow the instructions ,so considering P1 then P2 then P3, the only things that helped me is knowing that at the end of each P it’ll be facing parallel to the x axis, and that P4 is kind of a repetition/symmetry in a sense with P3 since P3LP3R however it still took me over 5 minutes to draw and prone to error

Reply 5

mqb2766

Its worth noting that "LR" terms cancel which simplifies the path a bit, but you can think of each segment being repeated with an appropriate turn at the end. With a bit of hindsight,
P2 = FLFFLFRR
P4 = P2 L P2 P2 L P2 RR
is a reasonable breakdown which is the 4 line "C" repeated 4 times with an appropriate turn at the end.

I guess it would take the 3-5 minutes to do it.