SOLUTION: Which terms of the sequence 2187,729,243.... Is 1÷9

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Question 1180646: Which terms of the sequence 2187,729,243.... Is 1÷9

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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This sequence is the geometric progression with the first term  2187 = 3%5E7  and the common ratio of  1%2F9.


To find the number of the term  1%2F9  in this progression,  write an equation for the general term of a GP

    a%5Bn%5D = a%5B1%5D%2Ar%5E%28n-1%29.


In your case this formula takes the form


    1%2F9 = 3%5E7%2A%281%2F3%29%5E%28n-1%29,   or


    1%2F3%5E2 = 3%5E%287-%28n-1%29%29.


From this equation


    -2 = 7 - (n-1)

    -2 = 7 - n + 1

    -2 = 8 - n

     n = 8 + 2 = 10.



ANSWER.  1%2F9  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.

Solved.

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On geometric progressions,  see introductory lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

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

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.