For example, when a detective puts together specific clues to solve a mystery.

A lot of what we will be working with in this lesson are sequences.  In general, a sequence is an ordered arrangement of numbers, figures, or objects. 

Specifically, sequences of math are a string of numbers that are tied together with some sort of consistent rule, or set of rules, that determines the next number in the sequence. 
 

The following are some specific types of sequences of math:

Arithmetic sequence:  a sequence such that each successive term is obtained from the previous term by addition or subtraction of a fixed number called a difference.  The sequence  4, 7, 10, 13, 16, ...   is an example of an arithmetic sequence.  The pattern is that we are always adding a fixed number of three to the previous term to get to the next term. Be careful that you don't think that every sequence that has a pattern in addition is arithmetic. It is arithmetic if you are always adding the SAME number each time. 

Geometric sequence:  a sequence such that each successive term is obtained from the previous term by multiplying by a fixed number called a ratio.  The sequence 5, 10, 20, 40, 80, ....  is an example of a geometric sequence.  The pattern is that we are always multiplying by a fixed number of 2 to the previous term to get to the next term.  Be careful that you don't think that every sequence that has a pattern in multiplication is geometric.  It is geometric if you are always multiplying by the SAME number each time. 

Fibonacci sequence:  a basic Fibonacci sequence is when two numbers are added together to get the next number in the sequence.   1, 1, 2, 3, 5, 8, 13, ....  is an example of a Fibonacci sequence where the starting numbers (or seeds) are 1 and 1, and we add the two previous numbers to get the next number in the sequence.
 

Note that not all sequences fit into the specific patterns that are described above.  Those are just the more common ones.  So as you look at patterns, look for those as a possibility, but if it doesn't fit one listed above, don't assume it doesn't have a pattern.

In general, when looking for a pattern start simple and then go from there.  For example, see if there is some pattern in adding, subtracting,  multiplying, or dividing.  Maybe you are always adding the same number to the previous term to get the new term.  Or maybe you are subtracting the next multiple of three from the previous number.  Or you are multiplying by a sequence of even numbers.  Perhaps, you are always adding or subtracting the two previous terms to get to the next one.  Exponential growth is another good pattern to look for.  Maybe you are always squaring or cubing the term number to get your result.  Also, don't forget that sometimes the pattern of a sequence is a combination of operations.  Maybe you have to multiply by 2 and then add 5 to get to the next number in a sequence or the output of a function.  If a problem seems like it is taking forever to work,  try a different approach - a different kind of sequence. 

Once you find your pattern, you can use it to find the next terms in the sequence.

Final Answer:
The pattern is to add the next even number.  The next three terms would have to be 35, 47, and 61, since 25+10 = 35,  35 +12 = 47, 47 +14 = 61. 

Final Answer:
The pattern is multiplying by the next negative integer.  The next three terms are -840, 5040, and -35280, since 168(-5) = -840, -840(-6) = 5040,  5040(-7) = -35280. 

Final Answer:
The pattern is we are subtracting by the next power of three.
The next three terms would be -20, -263, and -992, since 61 - 81 = -20, -20 - 243 = -263,  -263  - 729 = -992. 

As with the numeric patterns, this is not all the possible types of patterns involving figures.  However, it does give you a way to approach the problem.

Example 4:   Write the next three figures in the pattern

...

With all of that in mind, I believe the next three figures would be a square with a vertical line, then a circle, then a square with a horizontal line: 

Example 5:   Write the next two figures in the pattern

...

With all of that in mind, I believe the next two figures would be 

For example, if you were needing to find the area of a specific rectangle.  You would use the general formula for the area of the rectangle and apply it to the specific rectangle.

Here are some ideas that might help you approach a problem requiring deductive reasoning:
 

Example 6:   Use the statements below to answer the question that follows:

1.  All people wearing hats have blonde hair.
2.  Some of the people have red hair.
3.  All people who have blonde hair like hamburgers.
4.  People who have red hair like pizza.
5.  Keith has blonde hair.
 

Which of the following statements MUST be true?
a.   Keith likes hamburgers.
b.   Keith has red hair.
c.   Keith likes pizza.
d.   Keith is wearing a hat.

On deductive reasoning, you need to be a 100% sure.  There can’t be any doubt.

Since statement 3 says that ALL people who have blonde hair like hamburgers and Keith has blonde hair, then statement a, Keith likes hamburgers, is a 100% guarantee.

Example 7:   Jerry, Kevin, Todd and Mark all live on the first floor of an apartment complex.  One is a manager, one is a computer programmer, one is a singer, and the other is a teacher.  Use the statements below to answer the question that follows.

A.  Jerry and Todd eat lunch with the singer.
B.  Kevin and Mark carpool with the manager.
C.  Todd watches CSI with the manger and the singer.

Question: Which is the manager?

Statement A,  Jerry and Todd eat lunch with the singer, doesn’t let us definitively eliminate anyone from being the manager.

However, statement B,  Kevin and Mark carpool with the manager, eliminates Kevin and Mark from being the manager.  And statement C,  Todd watches CSI with the manger and the singer, eliminates Todd. 

The only one that could be (100%, without a doubt) the manager is Jerry.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.