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What is the common ratio of the geometric sequence 2, 6, 18, 54,...?

1 Answer
Feb 3, 2015

#3#

A geometric sequence has a common ratio, that is: the divider between any two nextdoor numbers:

You will see that #6//2=18//6=54//18=3#
Or in other words, we multiply by #3# to get to the next.
#2*3=6->6*3=18->18*3=54#

So we can predict that the next number will be #54*3=162#

If we call the first number #a# (in our case #2#) and the common ratio #r# (in our case #3#) then we can predict any number of the sequence. Term 10 will be #2# multiplied by #3# 9 (10-1) times.

In general
The #n#th term will be#=a.r^(n-1)#

Extra:
In most systems the 1st term is not counted in and called term-0.
The first 'real' term is the one after the first multiplication.

This changes the formula to #T_n=a_0.r^n#
(which is, in reality, the (n+1)th term).