Sequences and Series

Sequences and the nth term

A sequence is an ordered list of numbers. A series is what you get when you add them up. Two kinds appear on this syllabus, and telling them apart is the first thing to do in any question:

• an arithmetic progression adds the same amount each time — the common difference $d$;
• a geometric progression multiplies by the same amount each time — the common ratio $r$.

To decide which you are looking at, take differences between consecutive terms. If they are all equal it is arithmetic. If they are not, divide instead; if those are all equal it is geometric.

Arithmetic progressions

Here $a$ is the first term and $l$ is the last term. Use the second form of $S_n$ when you already know the last term, and the first form otherwise.

Two facts about a progression give you two equations, and $a$ and $d$ fall out of solving them simultaneously. That is what almost every AP question is, underneath.

Geometric progressions

The two forms of $S_n$ are the same expression with the top and bottom both multiplied by $-1$. Use the left one when $r < 1$ and the right one when $r > 1$, purely to avoid negatives.

The sum to infinity

Add up a geometric progression forever and one of two things happens. If the terms are shrinking, the total settles on a fixed number. If they are not, it grows without limit.

$|r| < 1$ means $-1 < r < 1$: the ratio may be negative, but its size must be under 1. A progression with $r = -\tfrac12$ has a sum to infinity; one with $r = -2$ does not.

Sigma notation

$\displaystyle\sum_{r=1}^{n}$ means "add up, letting $r$ run from 1 to $n$". So

$$\sum_{r=1}^{12} (3r - 2) = 1 + 4 + 7 + \cdots + 34$$

which is an arithmetic progression with $a = 1$, $d = 3$ and $n = 12$, giving $S_{12} = \tfrac{12}{2}\big[2 + 33\big] = 210$.

Counting the terms is where marks go. From $r = 1$ to $r = 12$ there are 12 terms; from $r = 5$ to $r = 12$ there are $12 - 5 + 1 = 8$.