Sequences (when rule is known)

A sequence is a set of numbers that follows some kind of rule or pattern.  Sometimes you have to figure out the rule that creates the sequence, other times the sequence is given to you.

Typically, sequence problems ask you to find a particular entry in the sequence.   The term "$n$th" uses "$n$" to mean "any given number," so that "$n$th" is any term (the 3rd, 8th, or 97th). When you are asked to find the $n$th term, you are being asked to find the rule that would give you any term in the sequence, if you plugged that number in for $n$ (so if you need to find the 13th term, you would plug in $n=13$

For sequence problems in which you are given the rule that determines each term, read the problem carefully and fill in the sequence according to the rule.  Fill it in until you get to the specified term.

These kinds of problems usually give you the first term, the pattern that the terms follow, and the term that you’re looking for.  Start with the first term and fill in the rest.

Example: If the first term is 80, and each term is 90% of the previous term, what is the 4th term?

To solve this problem, you start with the first term: $80$.

You then use the rule to find the next term: $80 \times .90 = 72$

Then use the rule to find subsequent terms until you get the the one that you want (in this case, the 4th).

$$\eqalign{80\times.90&=72\\72\times.90&=64.8\\64.8\times.90&=58.32\\58.32\times.90&=52.488}$$

The sequence is: 80, 72, 64.8, 58.32, 52.488

If you want the 4th term, your answer is 58.32

Other sequence problems do not give you the rule.  You have to look at the sequence and figure it out.

These types of problems are a bit more difficult.  Start with the first two terms and try to figure out how they are related.  Find a rule, then test to see if it also fits for the second to third term.  If it does, you have your rule and you can proceed.  If the rule you find between the first two numbers doesn't work to find the third number, try to find another rule that does. 

Example: If you are given the following sequence, what is the next term?  $18, 9, 0, -9...$

To solve this problem, look at the first two terms: 18 and 9.  How could they be related?  $18 \div 2=9$

Try the rule on the next two terms: $9\div 2 = 4.5  4.5 \neq 0$

The rule didn't work.  Lets try another rule: $18-9=9$

Let's try that rule on the next two terms: $9-9=0$

Perfect!  It works.  The rule for this sequence is take each term, then subtract 9 to find the next term.

Then use the rule to find subsequent terms until you get the the one that you want (in this case, the next or 5th term).

$-9-9=-18&

The sequence is: 18, 9, 0, -9, -18

If you want the "next" term, your answer is -18.

Overall, find the rule (either it's given to you or you have to find it) and apply it to find the term that the question is asking for.