Friday, October 12, 2018

[STEP3_2018_Q2] Sequence of Functions & Mathematical Induction

     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 undergraduate mathematics.  Some colleges and university departments may also require STEP.
     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.
     

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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