Pedagogical advice: Canonical sequences
Introduction
The purpose of this report is to collect types of sequences that students frequently pose in guess my robot, and comment on their mathematical structure. The analysis of these sequences is presented as background knowledge for the teacher. It is not our intention that it will be taught upfront. Rather, after a student comes across such a sequence, and tackles it, the teacher might want to follow up with investigation of the formal aspects.
an = p*an-1 +q
This is one of the most common forms of sequences used by students. Note the equivalent complementary form:
an = p*(an-1 +q) = p*an-1
+p*q
The closed form of this type of sequence seems a bit more
complicated:
an = p* an-1 + q =
p*(p* an-2 + q) + q =
p2* an-2 +p* q + q = …
pn-1* a1 + q* Σ (pi-2) =
pn-1* a1 + q*(1- pn-1)/(1-p)
Students approach sequences of this type intuitively, trying various values for X and Y. Interestingly, solving it formally is not as hard as you might expect. If we take three consecutive terms, we can write 2 formulas:
an = p*an-1 +q
an+1 = p*an +q
We do a bit of algebra:
q = an - p*an-1
q = an+1 - p*an
an - p*an-1 = an+1 - p*an
p*(an - an-1) = an+1 - an
Finally, we get:
p = (an+1 - an) / (an - an-1)
q= an+1 - an*p