Patterns in Numbers
Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers:
The branch of Mathematics that studies patterns in whole numbers is called number theory.
Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study. Table 1 shows some key number sequences that are studied in Mathematics.
Table 1: Examples of number sequences
| Number Sequence | Sequence Type |
|---|---|
| 1, 1, 1, 1, 1, 1, 1, ... | All 1’s |
| 1, 2, 3, 4, 5, 6, 7, ... | Counting numbers |
| 1, 3, 5, 7, 9, 11, 13, ... | |
| 2, 4, 6, 8, 10, 12, 14, ... | |
| 1, 3, 6, 10, 15, 21, 28, ... | |
| 1, 4, 9, 16, 25, 36, 49, ... | |
| 1, 8, 27, 64, 125, 216, ... | |
| 1, 2, 3, 5, 8, 13, 21, ... | Virahānka numbers |
| 1, 2, 4, 8, 16, 32, 64, ... | |
| 1, 3, 9, 27, 81, 243, 729, ... | Powers of 3 |
Figure it Out
(1) Can you recognise the pattern in each of the sequences in Table 1?
(a) 1, 1, 1, 1, 1, 1, 1, ... : All terms are
(c)1, 3, 5, 7, 9, 11, 13, ...: each term increases by
(e)1, 3, 6, 10, 15, 21, 28, ...: each term is the
(g)1, 8, 27, 64, 125, 216, ...:
(i)1, 2, 4, 8, 16, 32, 64, ...: Powers of
(2) Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
(a)Next Three Numbers:
(b)Next Three Numbers:
(c)Next Three Numbers:
(d)Next Three Numbers:
(e)Next Three Numbers:
(f)Next Three Numbers:
(g)Next Three Numbers:
(h)Next Three Numbers:
(i)Next Three Numbers:
(j)Next Three Numbers: