[S1_20151226NPSW] Finding the General Term of a Sequence (2)

Problem

Introduction

     The above was discussed in this previous article.  The earlier parts of the problem are easy.  The major sticking point is finding the formula for  S_n.  We solved that using factorisation and observation, which I feel is the best way.  But what if you cannot do that and you are desperate (for example, in an exam or test)?

     This article introduces Newton’s Method, which can be used as a back-up method, even though it is not in the regular syllabus.

Solution (Newton’s Method)

 

Remark

     Note that number sequences in “IQ tests” (with no problem contexts) have been debunked.  In our case here, the numbers do have a certain regularity arising from the pattern of dots.  In fact this is an arithmetic progression.  What we are calculating is the sum of an arithmetic progression.  However, Newton’s Method extends beyond arithmetic progressions.

Suitable Levels

* Lower School Mathematics

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

* GCE ‘A’ Levels H2 Mathematics (revision)

* other syllabuses that involve number patterns and sequences

* any precocious or independent learner who is interested

[MathEdn 20151118] Abuse of The Equal Sign and Maths IQ Puzzles Debunked

Question

Introduction

     This puzzle is making its rounds on the Internet.  The “solution” is easy enough.

          3 ×   6 = 18                  

          4 ×   8 = 32                  

          5 × 10 = 50                  

          6 × 12 = 72                  

          7 × 14 = 98                  

Notice that the numbers in the second column is always twice the corresponding number in the first column.  For the number 10, we have

         10 × 20 = 200

Critique 1

     A few people might get tricked by taking 10 × 16 = 160,  since 16 is the next even number after  14.  But that is not my gripe.  My issue is with the abuse of the “=” sign.  This sign stands for “equal” which means, equal (Surprise!  Surprise!).  Equal means the same, having the same value.  3  is not equal to  18.  So we should not write “3 = 18”, because that is not true.  Writing “3 ×   6 = 18” is OK and correct, because it makes sense and it is a true statement.  Mathematics is not a jumble of nonsensical symbols, although to some people it seems like it.  The symbols have meanings.  And these symbols should not be abused.  If you want to say “corresponds to”, then you might want to use an arrow e.g. “3  ¾® 18”.  This is keeping with the modern concept of a function, in which a value is assigned unambiguously to another number.  Using a function notation we can write things like “f(3) = 18”.

Critique 2

     Another problem with puzzles such as this is that, using Lagrange Interpolation or Newton Interpolation and the like, it is always possible to invent a function that hits the given first few numbers and then any number you like (even a “wrong” number).  For the above puzzle, the Lagrange method allows us to cook up a function like this:-

This function looks complicated, but if you note carefully, when you substitute the numbers 3, 4, 5, 6, 7 and 10, one of the algebraic fractions with  x  becomes equal to  1  and the rest of them become  0.  Thus, we easily see that  f(3) = 18,  f(4) = 32,  f(5) = 50,  f(6) = 72  and  f(7) = 98.  For  f(10),  I could actually have chosen any value I like, but I chose the number 42.  So the correct answer does not have to be 200.  There is actually no single correct answer, since you can make it any answer you like!  This trick can be done on all similar puzzles, and hence these “IQ” puzzles have now been effectively debunked!