Question 1136792
The sequence 2​, 6​, 18​, 54​, ... is geometric. Find
​(1) the common​ ratio,
<pre>
{{{matrix(1,7,
matrix(1,2,common,ratio),
""="",
matrix(1,2,second,term)/matrix(1,2,first,term),
""="",
6/2,
""="",
3)}}}

{{{matrix(1,7,
matrix(1,2,common,ratio),
""="",
matrix(1,2,third,term)/matrix(1,2,second,term),
""="",
18/6,
""="",
3)}}}

{{{matrix(1,7,
matrix(1,2,common,ratio),
""="",
matrix(1,2,fourth,term)/matrix(1,2,third,term),
""="",
54/18,
""="",
3)}}}

Since they are all the same, that shows that the sequence is
geometric, and that the common ratio, r=3.
</pre>(2) the ninth ​term,<pre>
2​, 6​, 18​, 54​, 54∙3=162, 162∙3=486, 486∙3=1458, 1458∙3=4374, 4374∙3=13122
</pre>
​(3) a recursive rule for the nth​ term,
<pre>
a<sub>1</sub> = 2, a<sub>n+1</sub> = 3∙a<sub>n</sub> for n≥1.
</pre>
​(4) an explicit rule for the nth term
<pre>
a<sub>n</sub> = a<sub>1</sub>∙r<sup>n-1</sup>

a<sub>n</sub> = 2∙3<sup>n-1</sup>

Edwin</pre>