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Geometric sequences and series

A geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio, r.

a_{n}=a_{n-1}\cdot r\;\: or\;\: a_{n}=a_{1}\cdot r^{n-1}


Write the first five terms of a geometric sequence in which a1=2 and r=3.

We use the first given formula:

\\ a_{1}=2\\ a_{2}=2\cdot 3=6\\ a_{3}=6\cdot 3=18\\ a_{4}=18\cdot 3=54\\ a_{5}=54\cdot 3=162\\

Just as with arithmetic series it is possible to find the sum of a geometric series. It is found by using one of the following formulas:

S_{n}=\frac{a_{1}-a_{1}\cdot r^{n}}{1-r}\; \; or\; \; S_{n}=\frac{a_{1}(1-r^{n})}{1-r}

Videolesson: use the formula for the sum of a geometric series to determine the sum when a1=4 and r=2 and we have 12 terms.

Next Class:  Sequences and series, Binomial theorem