Need to Know

Sequences & Series

The rules you must have ready for the exam.

How to use this: say each rule in your head, then tap Show me to check.

Sequences

6 rules

Rule 01

Term rule.

$T_n$ gives any term — substitute the position for $n$ ($T_1$ uses $n=1$).

Rule 02

Arithmetic sequence.

Goes up/down by a fixed amount. $a = T_1$, common difference $d = T_2 - T_1$.

Rule 03

General term.

$T_n = a + (n-1)d$. Multiply out for a rule for any term.

Rule 04

Which term?

Set $T_n$ equal to the value and solve for $n$.

Rule 05

Consecutive terms.

Equal gaps: $T_2 - T_1 = T_3 - T_2$.

Rule 06

Two given terms.

Write each with $T_n = a+(n-1)d$, subtract to get $d$, then find $a$.

3 rules

Rule 07

A series is a sum.

$S_n = T_1 + T_2 + \cdots + T_n$.

Rule 08

Sum formula.

$S_n = \dfrac{n}{2}\{2a + (n-1)d\}$.

Rule 09

Working backwards.

$a = S_1$, $\ T_2 = S_2 - S_1$, $\ d = T_2 - T_1$.

3 rules

Rule 10

Spotting it.

The second differences are constant.

Rule 11

The rule.

$T_n = an^2 + bn + c$, with $a = \dfrac{\text{2nd difference}}{2}$.

Rule 12

Finding b and c.

Use $T_1$ and $T_2$ to make two equations and solve.

Mathslive.ie · Sequences & Series · Need to Know