Linear number sequences
In a nutshell
A linear sequence of numbers consists of a series of numbers that follow a certain pattern. Each number is related to the next in some way and this relation is described by a rule.
Rules for number sequences
There are four rules that you need to know involving adding, taking away, multiplying and dividing.
Rule 1: Add or subtract by the same number
This is where the same number is added or subtracted to give the next term in the sequence.
Example 1
$\begin{array}{ccccccc}&&&+5&&+5&&+5&&+5&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&7&&13&&18&&23& \\&&&&&&&&&&&&\\ \hline\end{array}$
In this case, the rule is: "Add five to the previous number".
Rule 2: Multiply or divide by the same number
This is where the same number is multiplied by or divided by to give the next term in the sequence.
Example 2
$\begin{array}{ccccccc}&&&\div2&&\div2&&\div2&&\div2&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&400&&200&&100&&50&&25& \\&&&&&&&&&&&&\\ \hline\end{array}$
In this case, the rule is: "Divide the previous number by two".
Rule 3: Add or subtract by a changing number
This is where the number being added or subtracted changes by the same amount each time it is added/subtracted to a term.
Example 3
$\begin{array}{ccccccc}&&&&+1&&+1&&+1&&&\\&&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&&\\&&&+1&&+2&&+3&&+4&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&3&&5&&8&&12& \\&&&&&&&&&&&&\\ \hline\end{array}$
In this case, the rule is: "Add one extra to the previous number".
Rule 4: Add the previous two terms
This is where the previous two terms are added to give the next.
Example 4
$\begin{array}{ccccccc}&&&0+1&&1+1&&2+1&&3+2&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&1&&1&&2&&3&&5& \\&&&&&&&&&&&&\\ \hline\end{array}$
In this case, the rule is "Add the previous two terms"
Finding missing numbers in a sequence
Using the previous/next terms
You can use the rule you have found to find further numbers in the sequence.
Example 5
Find the missing term in the sequence.
Find the rule.
"Add three to the previous number"
Work out the missing term.
$14+3=\underline{17}$
Using a general rule
A quick way to find any number in the sequence is to work out a general rule. Usually the letter n is used in place of the term you want to find.
PROCEDURE
1.  Find the rule for the sequence. 
2.  Use the following:  In a sequence where you add the same number use that number multiplied by $n$.
 In a sequence where you subtract the same number use $$ that number multiplied by $n$.

3.  Look at the first term and $n=1$ to decide what should be taken away or added to the first part of your general rule. 
4.  Use your general rule to find any term in the sequence. 
Example 6
Find the 30th term in the sequence:
$\begin{array}{ccccccc}&&&+3&&+3&&+3&&+...&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&5&&8&&11&&...& \\&&&&&&&&&&&&\\ \hline\end{array}$
Find the rule.
"Add three to the previous term"
Multiply n by three.
$3n$
Substitute in one in place of n to find out the change needed.
$3\times1=3\\31=2$
Take one away from $3n$ to give the general rule.
$3n  1$
Replace n with $30$ to work out the thirtieth term.
$(3\times30)1 = \underline{89}$