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

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

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Answer 1 · 2015-02-03T21:40:13

$3$ $3$

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

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

So we can predict that the next number will be $54 \cdot 3 = 162$ $54*3=162$

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

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

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 $T_{n} = a_{0} . r^{n}$ $T_n=a_0.r^n$
(which is, in reality, the (n+1)th term).

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

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