Sequences

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

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

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

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

If the first differences change but the $\text{second}$ 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.