Number sequences

Your mother may say "Why count rotis? Just eat!" Yet she notices you ate two yesterday and three the day before, and plans tomorrow's dough. A batsman notices the bowler sends every third ball outside off stump, and gets ready. Both are spotting patterns. Mathematics is, in large part, the search for patterns, and for why they happen.

A number sequence is a list of numbers made by a rule. Your book's Table 1 shows ten famous ones. Some rules are simple:

• All 1's: 1, 1, 1, 1, …
• Counting numbers: 1, 2, 3, 4, 5, … (add 1 each time)
• Odd numbers: 1, 3, 5, 7, 9, … (start at 1, add 2)
• Even numbers: 2, 4, 6, 8, 10, … (start at 2, add 2)

A shortcut: to find the 11th odd number without writing them all, double 11 and take away 1: 2 × 11 − 1 = 21. The 15th even number is simply 2 × 15 = 30.

Others grow faster:

• Triangular numbers: 1, 3, 6, 10, 15, 21, 28, … The jumps grow by one each time: +2, +3, +4, +5, …
• Squares: 1, 4, 9, 16, 25, 36, 49, … (1 × 1, 2 × 2, 3 × 3, …)
• Cubes: 1, 8, 27, 64, 125, 216, … (1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, …)

And three more:

• Virahānka numbers: 1, 2, 3, 5, 8, 13, 21, … Each is the sum of the two before it: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. They are named after Virahānka, an Indian scholar who wrote about them more than a thousand years ago.
• Powers of 2: 1, 2, 4, 8, 16, 32, 64, … Each is double the one before.
• Powers of 3: 1, 3, 9, 27, 81, 243, 729, … Each is 3 times the one before.

Once you know the rule, you can always write the next number.