Question 1180646
.
<pre>

This sequence is the geometric progression with the first term  2187 = {{{3^7}}}  and the common ratio of  {{{1/9}}}.


To find the number of the term  {{{1/9}}}  in this progression,  write an equation for the general term of a GP

    {{{a[n]}}} = {{{a[1]*r^(n-1)}}}.


In your case this formula takes the form


    {{{1/9}}} = {{{3^7*(1/3)^(n-1)}}},   or


    {{{1/3^2}}} = {{{3^(7-(n-1))}}}.


From this equation


    -2 = 7 - (n-1)

    -2 = 7 - n + 1

    -2 = 8 - n

     n = 8 + 2 = 10.



<U>ANSWER</U>.  {{{1/9}}}  is the 10-th term in this geometric progression.


To check my answer, you may write this sequence of indexes from 7 to -2 inclusive

       7, 6, 5, 4, 3, 2, 1, 0, -1, -2

and count that the index (-2) is the 10-th member in this sequence.
</pre>

Solved.


---------------


On geometric progressions, &nbsp;see introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-geometric-progressions.lesson>Problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-geometric-progressions.lesson>Word problems on geometric progressions</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
<U>"Geometric progressions"</U>.


Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.