Arithmetic Sequences and Series

Starting point: Years 9 and 10

Number patterns

A sequence is a list of numbers in order. Each number is a term. The first term is $T_1$, the second is $T_2$, and so on.

To continue a pattern, find the rule. In 4, 7, 10, 13, ..., each term is 3 more than the last.

Finding a rule for the nth term

If the difference is always 3, the rule has $3n$ in it. Compare:

Each term is one more than $3n$, so $T_n=3n+1$. Then $T_{50}=151$.

Patterns and straight lines

Plot $n$ against $T_n$ and the points lie on a straight line. The common difference is the gradient. That is why arithmetic sequences are sometimes called linear sequences.

Quick summary

• An arithmetic sequence adds the same amount $d$ each step.
• $T_n=a+(n-1)d$, where $a$ is the first term.
• The graph of $T_n$ against $n$ is a straight line with gradient $d$.
• $S_n=\frac{n}{2}(a+l)=\frac{n}{2}[2a+(n-1)d]$.
• $\sum$ means add up the terms.

Quiz