Dear Tvtokyo and students,
In O-level Elementary Mathematics, you would have
been exposed to conjectures in the form of number patterns. Try to
recall some techniques your school has taught you.
In addition, one ought to be exposed to common
patterns (not explicitly taught or elaborated in detail in schools
unfortunately) which I will elaborate with specific examples:
Pattern 1: 1, 3, 5, 7, ..., 2n + 1 or 2n - 1, ...
This is an odd number sequence.
Pattern 2: 0, 2, 4, 6, ..., 2n, ...
This is an even number sequence.
Pattern 3: 1, 4, 7, 10, ..., 3n - 2, ...
This is an arithmetic sequence.
Pattern 4: -1, 1/2, -1/4, 1/8, ..., (-1)(-1/2)^{n - 1},
...
This is an alternating geometric sequence.
Pattern 5: 1, 4, 9, 16, 25, ..., n^2, ...
This is a sequence of perfect squares.
Pattern 6: 1, 2, 6, 24, 120, 720, ..., n!, ...
This is a sequence of factorials.
Pattern 7: 1/3, 3/5, 5/7, 7/9, 9/11, ..., (2n + 1)/(2n +
3) or (2n -1)/(2n + 1), ...
In this sequence, the numerators are odd numbers and
denominators are 2 more than numerators.
Pattern 8: 2, 2/3, 2/9, 2/27, 2/81, ..., 2 / 3^{n - 1},
...
In this sequence, the numerator is always a constant 2 and
denominators are powers of 3, with each power being 1 less than the
sequence number.
Pattern 9: 1, 3, 6, 10, 15, 21, 28, ..., 1/2 n(n + 1),
...
In this sequence:
1 is the sum of 1 term;
3 is the sum of 2 terms, i.e., 1 and 2;
6 is the sum of 3 terms, i.e., 1, 2 and 3;
10 is the sum of 4 terms, i.e., 1, 2, 3 and 4;
and so on...
Thanks.
Cheers,
Wen Shih