Geometric sequences
A
geometric sequence is a sequence of numbers each of which, after the first, is obtained by multiplying the preceding number a constant number called the
common rate.
a_{1}
a_{2}=a_{1}·r
a_{3}=a_{2}·r=a_{1}·r^{2}
a_{4}=a_{3}·r=a_{1}·r^{3}
...
a_{n}=a_{1}·r^{n1}
3, 6, 12, 24, 48,... is a geometric sequence because each term is obtained by multiplying the preceding number by 2.
To solve exercises using geometric sequences you need the following formula:
The nth term: a_{n}=a_{1}·r^{n1}
where:
a_{1} = the first term of the sequence
r
= common rate
n = number of terms
a_{n} = nth term
Given 27, 9, 3, 1, ... Find a_{n} and a_{8}
Using a_{n}=a_{1}·r^{n1
}
Given a geometric sequence with a
_{2}=10 and a
_{5}=80. Find a
_{n}.
a_{n}=a_{1}·r^{n1}, so we need to find a
_{1}.
To find it we use the next system of equations:
solving by substitution:
,
and a
_{1}=5.
That is a_{n}=5·2^{n1}
Find the
10 term of the geometric sequence where
the first terms are: 1,3,9,27,...
