[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

[AM_20151119] Anticipation and Bridging as Proof Tactics

Question

Introduction

     In a previous article, I have showed an “evil” tactic that can be used against “evil” questions.  Here is another “evil” trigonometric proof question.

     Many traditionalist school teachers insist on starting either from the LHS or the RHS, and working all the way to the other side.  Personally I do not mind any form of presentation as long as it is logical.  But not many students are able to do that.  People tend to fall into the trap of beginning a proof with the statement that they are supposed to prove in the first place.  This is called circular reasoning (or “begging the question” or petitio principii).  It is definitely a no-no.  So there is some advantage to sticking to the traditionalist mould.  The disadvantage is, of course, that it stifles creativity and this gives a misleading image of mathematics to the learner.

More “Evil” Tactics

     Now, if we do not want to “break the rules”, perhaps we can “bend the rules” a little.  On a piece of rough paper, or in your mind, secretly work from both sides and try to bridge them in the middle.  Let us compare

Think:  How are they similar?  How are they different?

As you can see, the LHS already has a preponderance of  cos 75°.  One of these  cos 75°  must somehow disappear.  The  RHS  has  4 sin 75°  which the LHS does not have.  So if we start from the LHS, we can use our magic wand [SV4] to create  4 sin 75°  out of thin air, remembering to divide by the same thing, so that the original value does not get changed.

Solution

Reflection

     Have you learned anything from this problem?  Let us review the heuristics used to help us solve this problem successfully.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Special Variants of Heuristics (Good for Trigonometric Proofs)

SV1   Work from both sides and try to bridge them in the middle [~ H04]

SV2   comparing (similarities/differences) what you have now with what you want [~ H05]

SV3   anticipate what will happen in the end and what you must do now [~ H05]

SV4   Magic Wand or create something out of nothing (无中生有) [~ H09]

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* IB SL & HL Mathematics (revision)

* other syllabuses that involve trigonometry

* just about anyone who is interested, really

[AM_20151119ISSD] Differences of Squares Hiding under Square Roots

Question

Introduction

     Here is another question involving surds.  As we know, surds are literally absurd, because they are irrational.  How to we do this one?

Strategy

Solution

Remark

     Reflect: What did you learn from solving this question?

     For another example of using the difference of squares formula, please look at this article.

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

* GCE ‘O’ Level Additional Mathematics (indices / surds)

* challenge for GCE ‘O’ Level “Elementary” Mathematics (indices)

* revision for IB Mathematics HL / SL

* other syllabuses that involve indices and/or surds
* any precocious or independent learner who is interested

[AM_20151117ISCP] Estimating a Crazy “Prosperous” surd to the Nearest Integer

Question

Introduction

     To the Chinese, the number  8  (八)  is considered to be auspicious, because it sounds like “prosper” (发) in the various Chinese languages/dialects.  But the LHS expression featured above seems too prosperous for comfort.  There is an explosion of  8s  coupled with eighth roots.  How to even handle that?

Strategy

     As usual, often one good tactic is to look for patterns or chunks.  [H04]  Can you see the sub-expressions containing the eighth roots (highlighted in green and pink)?

Do you notice any similarities between the two chunks?  Do you notice any difference(s)?

If we call the green chunk  a  and the pink chunk  b,  we can remove the roots by taking the eighth powers.  Then we get whole numbers, which are less complicated.

Solution

Remark

     The crux of the problem is the factorisation of  a⁸ – b⁸.  It is based on repeated application of the difference of squares    X² – Y² = (X + Y) (X – Y)  formula which schools expect students to know.
     For another example of using the difference of squares formula, please look at thisarticle.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* “IP Mathematics” so called
* challenge for students taking GCE ‘O’ Level Additional Mathematics
* other syllabuses that involve surds
* precocious kids who like to test themselves