[H2_20150512SSSTRR] Finding a Recurrence Relation for Terms in a Series

Question

Introduction

     This question pertains to the relationship between the partial sums of a series and its terms.  I am not sure if all the junior colleges teach this explicitly, but students are expected to know or be able to observe this relationship.  Let us follow our nose and focus on the first part first.

Reminders

     For the series  u₁ + u₂ + ¼ + u_n–1 + u_n + ¼  ,  the n^th partial sum

                    S_n = u₁ + u₂ + ¼ + u_n–1 + u_n

                 S_n–1 = u₁ + u₂ + ¼ + u_n–1

Taking the difference, we see that

                    u_n = S_n – S_n–1

Innocuous looking, this is actually a very powerful formula.  It is applicable to all sequences and series (not only for arithmetic and geometric series).  That means this formula can always be used!

     Another thing to note is that sequences  u_n  and partial sums  S_n  (which are themselves another sequence) behave like functions.  [In advanced mathematics, they are in fact defined as functions with domain as the positive integers.]  What this means is that  S_n-1  has the same formula as  S_n  except that  n  is replaced with  (n – 1). 

Solution

Checking

     Actually, the question setter forgot that the formula works for  n > 1. 

     OK, let us check whether the formula really works.  We know that  u₁ = 3.  Let us tabulate and compare the recursive formula with the explicit formula.  You can do this on a piece of rough paper.

n	recursive un = f(un–1)	explicit un = 3´2n–1
1	u1 = 3	3´21–1 =  3
2	u2 = 2´  3   = 6	3´22–1 =  6
3	u3 = 2´  6  = 12	3´23–1 = 12
4	u4 = 2´12 = 24	3´24–1 = 24
5	u5 = 2´24 = 48	3´25–1 = 48

Challenge

     What if the question wanted a recurrence relation for  S_n?

  

Heuristics Used

H04. Look for pattern(s)

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve sequences and series