Arithmetic Sequences

A sequence can be defined by a $\text{position-to-term}$ rule, such as $u_n=3n+2$, which gives any term directly: $u_{10}=32$.

Or by a $\text{recurrence relation}$, which gives each term from the one before, together with a starting value: $u_1=5$, $u_{n+1}=2u_n-1$ generates $5,9,17,33,\dots$

A sequence is $\text{increasing}$ if $u_{n+1}>u_n$ for all $n$, $\text{decreasing}$ if $u_{n+1}<u_n$ for all $n$, and $\text{periodic}$ if its terms repeat in a cycle — for example $u_{n+1}=\frac{1}{u_n}$ with $u_1=2$ gives $2,\frac{1}{2},2,\frac{1}{2},\dots$ with period 2.

An $\text{arithmetic sequence}$ has first term $a$ and $\text{common difference}$ $d$, so each term is $d$ more than the previous one: $u_{n+1}=u_n+d$.

Getting from the first term to the $n$th takes $n-1$ steps, each adding $d$, so $u_n=a+(n-1)d$.

Arithmetic sequences grow (or shrink) $\text{linearly}$: plotting $u_n$ against $n$ gives points on a straight line of gradient $d$.

Write the sum forwards and backwards: $S_n=a+(a+d)+\cdots +(a+(n-1)d)$ and $S_n=(a+(n-1)d)+\cdots +(a+d)+a$.

Adding the two lines term by term, every column totals $2a+(n-1)d$, and there are $n$ columns. So $2S_n=n(2a+(n-1)d)$.

If you know the last term $l=a+(n-1)d$, the same result is $S_n=\frac{n}{2}(a+l)$: the number of terms times the average of the first and last.

In applied questions, decide first what $a$, $d$ and $n$ represent, and whether the question wants a term (one value) or a sum (a running total).

When solving $S_n=k$ for $n$, you get a quadratic. The number of terms must be a positive integer, so reject negative or non-integer roots — or interpret them, if the question asks "when does the total first exceed...".