GEOMETRIC SEQUENCE
Geometric Sequences
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant. We can write a formula for the th term of a geometric sequence in the form
nth term
,where is the common ratio between successive terms.
For geometric sequences, the common ratio is r, and the first term a1 is often referred to simply as "a". Since we get the next term by multiplying by the common ratio, the value of a2 is just:
a2 = ar
Continuing, the third term is:
a3 = r(ar) = ar2
The fourth term is:
a4 = r(ar2) = ar3
At each stage, the common ratio was raised to a power that was one less than the index. Following this pattern, the n-th term an will have the form:
an = ar(n – 1)
EXAMPLE
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Find the tenth term and the n-th term of the following sequence:
, 1, 2, 4, 8,...
The first thing I have to do is figure out which type of sequence this is: arithmetic or geometric. I quickly see that the differences don't match; for instance, the difference of the second and first term is 2 – 1 = 1, but the difference of the third and second terms is 4 – 2 = 2. So this isn't an arithmetic sequence.
On the other hand, the ratios of successive terms are the same:
2 ÷ 1 = 2
4 ÷ 2 = 2
8 ÷ 4 = 2
