[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

[AM_20151130IAXS] A Motif for the Absolutely Absolute

Problem

Introduction

     This question would pose a challenge for many students, although theoretically it is within reach of a good Additional Mathematics student (~ grade 10).  Graphs of both  sin x  and  cos x  are waves that oscillate up and down.  There are many pairs of vertical bars, indicating the absolute values or modulus, and these seem confusing.

Strategy

     Let us graph the functions  y = |cos x|   and  y = |sin x|.   Note that  ||sin x| – |cos x|| = ||cos x| – |sin x||.   The absolute difference of  |cos x|   and  |sin x|  is the difference between them ignoring the negative sign (if any) of the result.  And this is just the difference between the higher value and the lower value. 

Can you see any repeating patterns?  [H04]  Can you visualise the required area?  How many times is that of the basic pattern (known as “motif” in art)?  [H09, H10, H11]

Solution

Remarks

     Our total area is made up of four congruent pieces.  When  0 < x < ^p/₄,  cos x  is higher than  sin x.  That allows us to strip away all the absolute signs and do the calculation.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* challenge for GCE ‘O’ Additional Mathematics  * IB Mathematics SL HL

* GCE ‘A’ Level H2 Mathematics  * IB Mathematics HL

* AP Calculus AB & BC

* University / College calculus

* other syllabuses that involve integration and area

* whoever is game for a challenge in integration

[PriCNPTGN_20150508] Sum of the First Few Natural Numbers

Question

 Introduction

     Numerical patterns are a challenge to learn and to teach.  Those of us who are teachers, usually the better students of our time, tend to think that the pattern is “obvious”, and hence do not bother to explain and/or facilitate classroom discussions regarding patterns.

     This article is about a pattern that involves the sum of the first few natural numbers (or wholenumbers).  This pattern, common in our primary school mathematics patterns, is a special case of the sum of an arithmetic progression, but its formula is not usually taught explicitly.  The story is told of the great mathematician Karl Friedrich Gauss, who figuredout a short-cut for adding up  1 + 2 +... + 100.  Lesser mortals in primary schools are left to struggle with frustration, or to copy “model answers” from their tutors or teachers without understanding how the solutions were obtained.
     As in a previous article, I attempt to illustrate the pattern visually.  I shall highlight the linkages to facilitate discovery of the general method, as well as show how the last part (part(c)) can be solved.

Solution

     Many pupils are able to deduce the answer to part (a) by analysing the differences between successive answers, which is equivalent to asking what must you add to get the next number.  For example, from the 1st number  1,  you add  2  to get the 2nd number  3,  you add  3  to get the third number  6.  To get the 4th number, you add  4  to get  10.  This approach works, but will not help you much for parts (b) and (c) of the question.  The better way is to look for a method that does not require you to keep on adding numbers.  That would allow one to kill all the birds with one stone.  How to do this?

     Imagine the given dots being doubled, rotated and then put together into parellelogram-like matrices.  If you want, you can imagine them as rectangular arrays.  I use the colour orange for the originals, and blue for the copies.

     As you can see, for figure 1, the total number of blue and orange dots is  1 ´ 2.  For figure 2, the total is  2 ´ 3.  For figure 3, the total is  3 ´ 4.  For figure 4, the total is  4 ´ 5.  The pattern is: for whatever number representing the position of the figure, the total is this ordinal number multiplied by another factor that is one more than this number.  Hence the number of original dots (shown in orange) is this product divided by  2.  With this insight, we can fill up the table to answer part (a).  The answer for figure  5  is  30.

We can also figure out that the answer to part (b) is  55,  with the above-mentioned pattern.

For part (c), we can use trial and error or “guess and check”.  Trying  21  gives  21´22 = 462  which does not work.  We try 22:  22´23 = 506.   Yes!  Bingo!

You may also use a calculator to help you.  Since the two unknown factors are close together (they differ only by one), it is almost like multiplying a number with itself, or squaring.  So to guess our number, we may use the square root (the opposite of squaring) to estimate it.  The square root of  506  is about  22.49.  We guess  22  and verify that 22´23 = 506.

Summary

     The pattern involving the sum of the first few whole numbers may be deduced by making a copy of the original figure, rotating it and joining it to form an array.  Just multiply accordingly and then divide by  2  to get the sum for the original figure.  To solve for a figure’s ordinal number (which figure has a certain given number of dots), one may use guess and check, or use square roots.  A primary school pupil should be able to all these without the knowledge of advanced techniques like the sum of arithmetic progressions or quadratic equations.

Heuristics Used

H02. Use a diagram / model

H03. Make a systematic list

H04. Look for pattern(s)

H05. Work backwards

H07. Use guess and check

H09. Restate the problem in another way

Suitable Levels

* Primary School Mathematics (algebra unnecessary)

* GCE ‘O’ Level “Elementary” Mathematics (Number patterns, with algebra)

* GCE ‘A’ Levels H2 Mathematics (Number patterns, with algebra)
* anyone who loves a challenge to unravel a pattern

[S2_ABNPQX_20150502] Jack-in-the-Box of Fractions

Question

Introduction

     This number pattern question requires a lot of observation and algebra skills.  One also needs to understand how subscript notation like  F_n  works.

Solution