[H2_20150514GAS] Grouped Arithmetic Sequences

Question

Introduction

     This question, even though not the most difficult of its type, poses quite a challenge to many students.  Without the curly brackets, the sequence is just an ordinary arithmetic progression (AP) or arithmetic sequence.  Once the curly brackets are in, it messes up our mental schema.  We can no longer use the formulas for AP naïvely.

     Actually, we should never apply any mathematical formula blindly.  Neither we should be stuck with a literal interpretation of the symbols.  We should apply formulas according to their meaning.  For example, in the sum-of-an-AP formula                                       

the  n  represents the number of terms.  But the  n  as used in the question has a different meaning.  It means the set number.  The best students are able to observe this, and hold the difference in meanings in their heads when they apply the formulas.  This requires some mental effort.  It is easy to make a careless mistake if you lose your focus or concentration.  If you are not so confident of doing this mentally, and you want to play it safe, I would suggest that you use another set of symbols (say, capital letters)                                        .

You can also use subscript notations like  A_n  for the first member in set  n.

Observations

     Before trying to do anything.  It is always good to take a step back and make observations, and play with small numbers first.  Once you have observe the patterns, you can plan your strategy to tackle the question.

Solution

Remarks

     As you can see, the actual presentation of the solution is actually quite short.  But there is a lot of thinking behind it.  It is important to make observations, even if some of them seem unnecessary for this question.  This allows you to solve problems even more challenging than this.  For example, what if the bare sequence did not start from  1  and has a common difference more than  1? 

          {5},  {8,  11},  {14, 17, 20},  {23, 26, 29, 32},  ...

What if the bare sequence was a geometric progression?  Like this

          {1},  {2,  4},  {8, 16, 32},  {64, 128, 256, 512},  ...

What if the bare sequence was an arithmetic progression, but the number of terms in the sets follow a geometric progression?  Like this

          {3},  {5,  7},  {9, 11, 13, 15},  {17, 19, 21, 23,  25, 27, 29, 31},  ...

Happy figuring these out!

HeuristicsUsed

H04. Look for pattern(s)

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve arithmetic and geometric progressions