Q1.
Can you describe the pattern in each of the above sequences? Can you predict the next few numbers in these sequences?
Answer
Each of the four sequences is described by what happens from one term to the next, and that alone fixes every later term.
| Sequence | Pattern in the gaps | Next three terms | Rule for the nth term |
|---|---|---|---|
| 1, 2, 3, 4, 5, 6, … | add 1 each time | 7, 8, 9 | tn = n |
| 1, 3, 5, 7, 9, 11, … | add 2 each time | 13, 15, 17 | tn = 2n – 1 |
| 1, 3, 6, 10, 15, 21, … | gaps are 2, 3, 4, 5, 6 — they grow by 1 | 28, 36, 45 | tn = n(n + 1)/2 |
| 1, 4, 9, 16, 25, 36, … | gaps are 3, 5, 7, 9, 11 — the odd numbers | 49, 64, 81 | tn = n2 |
Check the last two rows against the printed terms: 21 + 7 = 28, 28 + 8 = 36, 36 + 9 = 45; and 36 + 13 = 49, 49 + 15 = 64, 64 + 17 = 81. The gaps themselves form a sequence, and that is what makes prediction possible.
Why it happens: the triangular numbers are the running totals of 1, 2, 3, 4, …, so the gap between the (n – 1)th and the nth is exactly n. The square numbers are the running totals of 1, 3, 5, 7, …, so the gap there is the nth odd number, 2n – 1. In both cases you are not guessing — you are reading off a structure the sequence already has.
Check it yourself: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28, and 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49. The two lists of running totals really do give the triangular and square numbers.