12.2. Arithmetic and Geometric Sequences http://www.ck12.org
a 1 =a 1
a 2 =a 1 +d
a 3 =a 1 + 2 d
a 4 =a 1 + 3 d
...
an=a 1 +(n− 1 )dGeometric sequences are defined by an initial valuea 1 and a common ratior.
a 1 =a 1
a 2 =a 1 ·r
a 3 =a 1 ·r^2
a 4 =a 1 ·r^3
...
an=a 1 ·rn−^1If a sequence does not have a common ratio or a common difference, it is neither an arithmetic or a geometric
sequence. You should still try to figure out the pattern and come up with a formula that describes it.
Example A
For each of the following three sequences, determine if it is arithmetic, geometric or neither.
a. 0. 135 , 0. 189 , 0. 243 , 0. 297 ,...
b.^29 ,^16 ,^18 ,...
c. 0. 54 , 1. 08 , 3. 24 ,...Solution:
a. The sequence is arithmetic because the common difference is 0.054.
b. The sequence is geometric because the common ratio is^34.
c. The sequence is not arithmetic because the differences between consecutive terms are 0.54 and 2.16 which are
not common. The sequence is not geometric because the ratios between consecutive terms are 2 and 3 which
are not common.Example B
For the following sequence, determine the common ratio or difference, the next three terms, and the 2013thterm.
(^23) , (^53) , (^83) , (^113) ,...
Solution:The sequence is arithmetic because the difference is exactly 1 between consecutive terms. The next three
terms are^143 ,^173 ,^203. An equation for this sequence would is:
an=^23 +(n− 1 )· 1
Therefore, the 2013thterm requires 2012 times the common difference added to the first term.
a 2013 =^23 + 2012 · 1 =^23 +^60363 =^60383