This is a challenging problem on a sequence of functions, appearing as a
Sixth
Term Examination Papers (STEP) question.
STEP is the entrance exam for University
of Cambridge and the University of Warwick
The student does not need to know that the question involves a Rodrigues
type of formula. However great facility
in symbolic manipulation including algebra and calculus is needed, as this is
what would be expected of students in a rigorous course involving mathematics
or a related discipline.
The first part is done via Differentiation using the Product Rule and
the Chain Rule.
For the mathematical induction proof in part (ii), the following is a
rather standard way to begin. You should
do this even if you do not feel confident about the proof. Just write it down, and worry later. Say something like “RTP” (required to prove)
or “to be proven” so as not to give the impression that you are making unproven
assertions or making circular arguments, like what modern journalists and
political activists are prone to do.
The starting case is usually easier to handle. Just follow your nose and differentiate using
the Product Rule and the Chain Rule with
n = 1.
The next part, the induction step, is the
most challenging part. The trick is to
be clear about what is required and be observant. There are no derivatives in the final
required expression, and yet you should know that the earlier part of the
question serves as a hint that you must use derivatives. Using the induction hypothesis [IH], we end
up with an expression that has two derivatives, which is a pain to do by hand. So we repeatedly make use of [1] to convert
back to some expression involving the function sequence, but not involving
derivatives. After some cancellation and
simplification we finally complete the step.
The following is the standard type of conclusion for mathematical
induction proofs. Just remember to write
it in and earn the marks allocated.
The last part is again challenging.
The key to solving it is to observe that
‘x’ does not appear explicitly in the desired
final expression. So we proceed to try
to eliminate the term that contains ‘x’.
Examining the LHS would suggest the types of terms that we need formulas
for, and upon subtraction, will kill off the term that contains ‘x’.
Ta da! Done finally!
To recap: the strategies used to
solve this question is observation, anticipation (know what
you want at the ‘end of the rainbow’) and elimination (get rid of the unwanted
term). Needless to say, you would also
need to be thoroughly familar with the standard ‘A’ level Further Maths stuff
involving differentiation using the Product Rule and the Chain Rule, sequences
and mathematical induction.
This article is suitable for
* GCE
‘A’ Level Further Mathematics
* Students doing STEP and/or students applying to study
undergraduate mathematics in Cambridge / Oxford / Warwick
* other syllabuses calculus and sequences
* any learner who is interested
When we know for sure some pattern always repeats itself, we can use it reliably.
ReplyDeleteTo the ancient Greeks, talking about philosophy (including logic, ethics, morality, ...) was considered practical. The "how to do it" follows from there. Today, we do it the other way round. That is why our world (including business, politics, environment, geopolitics ... etc) is all messed up.