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# 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.

a1
a2=a1·r
a3=a2·r=a1·r2
a4=a3·r=a1·r3
...
an=a1·rn-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 nth term: an=a1·rn-1
where:
a1 = the first term of the sequence
r = common rate
n = number of terms
an = nth term

Given 27, -9, 3, -1, ... Find an and a8

Using
an=a1·rn-1  Given a geometric sequence with a2=-10 and a5=-80. Find an.

an=a1·rn-1
, so we need to find a1.

To find it we use the next system of equations: solving by substitution: , and a1=-5.

That is an=-5·2n-1

 Find the 9 term of the geometric sequence where 1 = 3 and r = 3

Solution =