**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*for the first member in set_{n}*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!

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

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