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 a
2=-10 and a
5=-80. Find a
n.
an=a1·rn-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 an=-5·2n-1
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Find the 11 term of the geometric sequence where the first terms are: -4,-12,-36,-108,...
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