You drop a rubber ball from a height of 48 inches. Each time it bounces, it reaches lower and lower heights. The following sequence shows its height with each successive bounce. How high will the ball bounce on its fifth bounce?
48, 36, 27, 20.25,...
Guidance
When looking at a sequence of numbers, consider the following possibilities.
 There could be a common difference (the same value is added or subtracted) to progress from each term to the next.
Example:
 There could be a common ratio (factor by which each term is multiplied) to progress from one term to the next.
Example:
 If the terms are fractions, perhaps there is a pattern in the numerator and a different pattern in the denominators.
Example:
 If the terms are growing rapidly, perhaps the difference between the term values is increasing by some constant factor.
Example:
 The terms may represent a particular type of number such as prime numbers, perfect squares, cubes, etc.
Example:
 Consider whether each term is the result of performing an operation on the two prior terms.
Example:
 Consider the possibility that the value is connected to the term number:
Example:
In this example
This list is not intended to be a comprehensive list of all possible patterns that may be present in a sequence but they are a good place to start when looking for a pattern.
Example A
Find the next two terms in the sequence:
Solution:
Each term is the result of multiplying the previous term by
Example B
Find the next two terms in the sequence:
Solution:
The difference between the first two terms
Example C
Find the next two terms in the sequence:
Solution:
This sequence requires that we look at the previous two terms. To get the third term, the second term was subtracted from the first:
Intro Problem Revisit
Each successive term in the sequence is the result of multiplying the previous term by
Therefore, the ball reaches a height of 15.1875 inches on its fifth bounce.
Guided Practice
Find the next two terms in each of the following sequences:
1.
2.
3.
Answers
1. Each term is the previous term plus 4. Therefore, the next two terms are 11 and 15.
2. The pattern here is somewhat hidden because some of the fractions have been reduced. If we “unreduced” the second and fourth terms we get the sequence:
3. This sequence is the set of perfect squares or the term number squared. Therefore the
Explore More
Find the next three terms in each sequence.

15,21,27,33,−−−−,−−−−,−−−− 
−4,12,−36,108,−−−−,−−−−,−−−− 
51,47,43,39,−−−−,−−−−,−−−− 
100,10,1,0.1,−−−−,−−−−,−−−− 
1,2,4,8,−−−−,−−−−,−−−− 
72,53,34,15,−−−−,−−−−,−−−−
Find the missing terms in the sequences.

1,4,−−−−,16,25,−−−− 
23,34,−−−−,56,−−−− 
0,2,−−−−,9,14,−−−− 
1,−−−−,27,64,125,−−−− 
5,−−−−,11,17,28,−−−−,73 
3,8,−−−−,24,−−−−,48 
1,1,2,−−−−,5,−−−−,13  Do any of the problems above have a constant difference? If so, which ones and what is the constant?
 Do any of the problems above have a common ratio? If so, which ones and what is the ratio?