Sequences

A $\textbf{term-to-term}$ rule tells you how to get the next term: "start at 4, add 3" gives $4, 7, 10, 13, \dots$

A $\textbf{position-to-term}$ rule, or $n$th term, gives the value from its position $n$: $3n + 1$ gives $4, 7, 10, 13, \dots$ for $n = 1, 2, 3, 4$.

Special sequences to recognise: square numbers $1, 4, 9, 16$; cube numbers $1, 8, 27, 64$; triangular numbers $1, 3, 6, 10$.

A linear (arithmetic) sequence has a constant difference $d$. Its $n$th term is $dn + b$, where $b$ is found by comparing with the sequence $dn$.

For $7, 11, 15, 19$: the difference is 4, so start with $4n$ ($4, 8, 12, 16$). Each term is 3 more, so the $n$th term is $4n + 3$.

To test whether a number is in the sequence, set the $n$th term equal to it and solve. A whole-number $n$ means yes.

A $\textbf{geometric}$ sequence multiplies by the same number $r$ each time: $5, 15, 45, 135, \dots$ has $r = 3$. Its $n$th term is $ar^{n-1}$, here $5 \times 3^{n - 1}$.

A $\textbf{Fibonacci-type}$ sequence adds the two previous terms: starting $2, 5$ gives $7, 12, 19, 31, \dots$

Geometric ratios can be fractions or surds: $\sqrt2, 2, 2\sqrt2, 4$ has ratio $\sqrt2$.

If the first differences change but the $\textbf{second}$ differences are constant, the sequence is quadratic.

The coefficient of $n^{2}$ is half the second difference. Subtract that $n^{2}$ term from the sequence, and what remains is linear — find its $n$th term as usual.