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Arithmetic and geometric sequences
Geometric series
A geometric series is the sum of a geometric sequence. That is,

a + ar + ar2 + ar3 + ar4 + ...

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

3 + 6 + 12 + 24 + 48 +... is a geometric serie because each term is obtained by multiplying the preceding number by 2.

To solve exercises using geometric sequences you need the following formulas:
• The nth term: an=a1·rn-1
• The sum of the first n terms:

• The sum of an infinite geometric serie: only if |r|<1.

where:
a1 = the first term of the sequence
r = common rate
n = number of terms
an = nth term
Sn = sum of the first n terms
S= sum of an infinite geometric serie

Given $\frac{1}{6}+\frac{1}{2}+\frac{3}{2}+...$, find S7
This is a geometric serie with a common rate r=3.

 Determine the sum of the first 7 terms of the geometric sequence where 1 = -5 and r = 3

Solution =