[S1_20151226NPGT] Finding the General Term of a Sequence (1)

Problem

Introduction

     This is a typical Secondary 1 type of problem involving number patterns.  Students are usually able to see the link between successive terms, but the general formula seems to be a challenge for most.

Strategy

     In case this is not obvious, every time you go to the next diagram, you add four dots on the outside.  So you can fill in the table very easily.  For diagram 5, there would be  19  dots and the total number of dots up to diagram  5  would be  55.

     What is the number of dots for diagram  1 000  or any number  n  for that matter?  Now, some students may have a problem predicting beyond the first few numbers.  What we need is a expression or formula that predicts the number of dots given the diagram number  n.  You know that the sequence  3,  7,  11,  15,  ...  follow a pattern where you keep adding  4.  Have you encountered a sequence in which  4  is added each time?  Yes!  It is the 4 times table.  Suppose we have the 4 times table.  [H08]   Let us do a comparison between that and  D_n.

diagram #	1	2	3	4	5	...	n
4 times table	4	8	12	16	20	...	4n
Dn	3	7	11	15	19	...	?

The numbers in  D_n  are always one less than those in the  4  times table.  So  D_n = 4n – 1.

Solution

n	number of dots for the  nth  diagrams Dn	Sum of number of dots for the first  n  diagrams Sn
1	3	3
2	7	10
3	11	21
4	15	36
5	19	55

     D_n = 4n – 1

     S_n  = n(2n + 1)   ©

Commentary

     How can we get the formula for  S_n?  We can do so by trying to factorise the numbers [H09], and then look for pattern.  [H04, H05]

           3 = 1×3   = 1×(2×1+1)

         10 = 2×5   = 2×(2×2+1)

         21 = 3×7   = 3×(2×3+1)

         36 = 4×9   = 4×(2×4+1)

         55 = 5×11 = 5×(2×5+1)

         ...

                      S_n = n(2n + 1)   ©  bingo!

But what if you have poor observational powers and if you are desperate?  There is a secret weapon to handle this!  Please refer to this article.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H08. Make suppositions

H09. Restate the problem in another way

Suitable Levels

* Primary School Mathematics (challenge)

* Lower Secondary Mathematics (Sec 1 ~ grade 7)

* GCE ‘O’ Level “Elementary” Mathematics (revision)

* other syllabuses that involve number patterns and algebra

* any precocious or independent learner who loves number patterns