http://www.ck12.org Chapter 12. Discrete Math
Example C
For the following sequence, determine the common ratio or difference and the next three terms.
(^23) , (^49) , 276 , 818 , 24310 ,...
Solution:This sequence is neither arithmetic nor geometric. The differences between the first few terms are
−^29 ,−^29 ,−^1081 ,− 24314. While there was a common difference at first, this difference did not hold through the se-
quence. Always check the sequence in multiple places to make sure that the common difference holds up
throughout.
The sequence is also not geometric because the ratios between the first few terms are^23 ,^12 ,^49. These ratios are not
common.
Even though you cannot get a common ratio or a common difference, it is still possible to produce the next three
terms in the sequence by noticing the numerator is an arithmetic sequence with starting term of 2 and a common
difference of 2. The denominators are a geometric sequence with an initial term of 3 and a common ratio of 3. The
next three terms are:
(^1236) , (^1437) , (^1638)
Concept Problem Revisited
Example C shows that some patterns that use elements from both arithmetic and geometric series are neither
arithmetic nor geometric. Two famous sequences that are neither arithmetic nor geometric are the Fibonacci
sequence and the sequence of prime numbers.
Fibonacci Sequence: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 ,...
Prime Numbers: 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 ,...
Vocabulary
Asequenceis a list of numbers separated by commas.
The common pattern in anarithmetic sequenceis that the same number is added or subtracted to each number to
produce the next number. This is called thecommon difference.
The common pattern in ageometric sequenceis that the same number is multiplied or divided to each number to
produce the next number. This is called thecommon ratio.
Guided Practice
- What is the tenth term in the following sequence?
− 12 , 6 ,− 3 ,^32 ,... - What is the tenth term in the following sequence?
− 1 ,^23 ,^73 , 4 ,^173 ,... - Find an equation that defines theakterm for the following sequence.
0 , 3 , 8 , 15 , 24 , 35 ,...
Answers: - The sequence is geometric and the common ratio is−^12. The equation isan=− 12 ·(−^12 )n−^1. The tenth term is:
− 12 ·(−^12 )^9 = 1283
- The pattern might not be immediately recognizable, but try ignoring the^13 in each number to see the pattern a