Decreasing sequences and their rules — Mathematics, 11–13
In a decreasing sequence every term is smaller than the one before, and the rule shows up when you look at how each term comes from the previous one or from its position.
Each term smaller than the last
A sequence is decreasing when each term is smaller than the one before. To find the rule, ask how each term is made from the previous one. In 90, 78, 66, 54… you subtract 12 each time, so the next term is 42. In 1/3, 1/6, 1/9, 1/12… the numerator stays 1 and the denominators are 3, 6, 9, 12: in position n the term is 1/(3 × n). In 2/3, 2/9, 2/27, 2/81… the denominators are the powers 3, 9, 27, 81, so in position n the term is 2/3ⁿ. A rule can be said in words or in symbols, and it stays a conjecture until it fits every term you know.
Skipping ahead, filling gaps, knowing the end
To find the 12th term of 200, 185, 170… you can subtract 15 eleven times, or do one calculation from the position: 200 − 15 × 11 = 35. Comparing the two ways is part of solving problems: listing terms is safe but slow; a rule from the position is fast but must be tested. A rule also fills a gap (48, 40, ?, 24 gives 32) and lets you create a sequence: pick a first term and a rule, such as start at 100 and subtract 7. And it shows how a sequence ends: 200 − 15 × 13 = 5, and the next subtraction would go below 0, while 2/3, 2/9, 2/27… never reach 0.
The 10th term, and is 50 a term?
Sequence 200, 185, 170, 155… (subtract 15). The 10th term is 9 jumps after the first, so 200 − 15 × 9 = 200 − 135 = 65. Is 50 a term? From 200 down to 50 is 150, and 150 ÷ 15 = 10 jumps exactly, so 50 is the term in position 11. Is 100 a term? From 200 down to 100 is 100, and 100 ÷ 15 = 6 remainder 10, not a whole number of jumps: after 6 jumps you are at 110, after 7 at 95, and 100 is skipped. So 100 is not a term.
Position n has n − 1 jumps
Sequence 20, 17, 14, 11… Many people write the term in position n as 20 − 3 × n: 20 is the start and 3 is taken away each time. It sounds right, but test it: for n = 1 it gives 20 − 3 = 17, and the first term is 20. The first term has made no jump yet, so the term in position n has n − 1 jumps: 20 − 3 × (n − 1), the same as 23 − 3 × n. Check: n = 1 gives 20, and n = 4 gives 23 − 12 = 11. Always test a rule on the terms you already have.
Countdowns, tanks and paper sizes
Decreasing sequences appear in a countdown (10, 9, 8…) and in a tank that loses the same amount of water each day. Paper sizes give one with fractions: a sheet of size A0 has an area of 1 m², and each next size has half the area of the one before. So A1, A2, A3 and A4 have 1/2, 1/4, 1/8 and 1/16 of a square metre, the denominators being powers of 2. The usual sheet, A4, is 1/16 m², the same kind of pattern as 2/3, 2/9, 2/27…
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