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^{n-1}

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 n th term: a_{n} =a_{1} ·r^{n-1}

where:

a_{1} = the first term of the sequence
r
= common rate
n = number of terms
a_{n} = n th term

Given 27, -9, 3, -1, ... Find a_{n} and a_{8}
Using a_{n} =a_{1} ·r^{n-1
}

Given a geometric sequence with a

_{2} =-10 and a

_{5} =-80. Find a

_{n} .

a_{n} =a_{1} ·r^{n-1} , 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^{n-1}

Find the
8 term of the geometric sequence where
_{3} = 45 and _{5} = 405