Number Patterns

1, 1, 2, 3, 5, 8, 13... what comes next? One of mathematicians' favorite games is flushing the hidden rule out of a run of numbers. Numbers lined up by a rule form a sequence, and each number in it is a term. Once you catch the rule, term 10 and term 100 both fall to your prediction.

The secret between neighbors

The most direct hunt: stare at neighboring terms and ask, "what happened from one to the next?"

Add 3 each time: 2, 5, 8, 11, 14... next comes 17;
Multiply by 2 each time: 1, 2, 4, 8, 16... next comes 32;
Take turns: +2, ×2, +2, ×2... gives 1, 3, 6, 8, 16 — the next move is an add, landing on 18.

Rules can add, multiply, or alternate shifts — so look at several terms before committing.

Look before you leap

1, 2, 4, 8 looks like "double each time", promising 16 next. But what if the continuation were 31? Then it would be the opening of the circle-splitting sequence. Verify with the terms you are given, and never bet on the first pattern you see.

InteractivePattern Machine

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

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This machine hoards rules: feed it a number, it returns a number, and the rule stays secret until you guess it. A sequence works the same way — the input is "which term", the output is "that term's value", and the invisible machine between them is the rule.

Term n: straight to the answer

"Add 3 each time" works, but reaching term 100 means adding 99 times. The smarter question: can term n be computed straight from n?

$n$	1	2	3	4	5
Term	4	7	10	13	16

The table's rule is "add 3 each time", which also reads $4 + 3 \times (n - 1)$ — in other words, $3n + 1$ . Verify twice: $n = 1$ gives 4, $n = 4$ gives 13, both match. Term 100 is now one step away: $3 \times 100 + 1 = 301$ .

A whole family of position-to-term formulas:

even numbers 2, 4, 6, 8... term n is $2n$ ;
3, 5, 7, 9... term n is $2n + 1$ ;
square numbers 1, 4, 9, 16... term n is $n^2$ .

Verify with two values

After guessing a formula for term n, substitute at least two values: $3n + 1$ gives 4 at $n = 1$ and 13 at $n = 4$ . One match may be a coincidence; two matches mean it is almost surely the rule.

Arithmetic sequences: equal strides

When every term exceeds the previous one by the same number, the sequence is called arithmetic, and that fixed step is the common difference. In 4, 7, 10, 13... the difference is 3.

A universal recipe for term n: first term + (n − 1) × difference. Term 10 of the sequence above:

$4 + 9 \times 3 = 31$

From term 1 to term 10 takes 9 strides — not 10. That off-by-one is the classic trap.

Sequences for fun

Triangular numbers: 1, 3, 6, 10, 15... each one arranges dots into a triangle, and term n is $\frac{n(n+1)}{2}$ ;
Look-and-say: 1, 11, 21, 1211, 111221... each term "reads" the previous one aloud: one 1, two 1s, one 2 one 1. It only counts if you say it out loud.

Rules do not always live in adding and multiplying — sometimes they hide in shapes, and sometimes in your own mouth. To feed the machine and guess rules yourself, play at Variables & the Function Machine.

Check yourself

Quick quiz

1. What comes next in 3, 6, 12, 24?
2. What is term 10 of the arithmetic sequence 5, 9, 13, ...?
3. Which formula gives term n of 3, 5, 7, 9?