A linear sequence goes up (or down) by the same amount each step — that constant gap is the common difference. A term-to-term rule says "add 2 to get the next one", but to reach, say, the 100th term you would have to add 2 again and again. The nth term rule fixes that: it is a single expression in the position n that you can substitute into to jump straight to any term.

Take the sequence 3, 5, 7, 9, \dots Each term is 2 more than the last, so the common difference is 2. That difference is the coefficient of n: the rule starts as 2n.

But 2n gives 2, 4, 6, 8, \dots — the 2 times table — which is 1 below every term we want. So we adjust the constant: add 1.

Check it: the 4th term is 2 \times 4 + 1 = 9. And the 100th term is 2 \times 100 + 1 = 201 — no counting required.

See it built

Watch the rule come out of the table: line up each position n against its term, read the common difference as the coefficient, then nudge the constant until the rows match. Step through it.