Sequences: nth term and arithmetic series

Key learnings

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  Linear nth term: dn + (first term − d)

  For a sequence going up by a fixed amount $d$, the $n$th term is $dn$ plus a correction, e.g. 5, 8, 11, … has $d=3$ and nth term $3n+2$.

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  Quadratic nth term: half the second difference is the n² coefficient

  If the second differences are constant and equal $2a$, the $n$th term starts with $an^2$; subtract $an^2$ from each term and find the linear part of what is left.

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  Arithmetic series sum

  $S_n=\dfrac{n}{2}\big[2a+(n-1)d\big]=\dfrac{n}{2}(\text{first}+\text{last})$, where $a$ is the first term and $d$ the common difference.