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

D_{n}

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
D_{n}

Sum of number of
dots for the first n
diagrams
S_{n}

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.
H04. Look for pattern(s)
H05. Work backwards
H08. Make suppositions
H09. Restate the problem in another
way
Suitable Levels
* 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
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