Question 1197284
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Find the 12th term of a Geometric Sequence whose 3rd term is 432 and the 5th term is 15552.
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            @MathLover1 wrote many symbols, but failed to solve the problem in a right way: 

            she missed one of the two solutions.


            So I came to bring a correct solution and correct answer.



<pre>
Since the 3rd and the 5th terms of the geometric progression are given, we write

    {{{a[5]}}} = {{{a[3]*r^2}}},

or

    15552 = {{{432*r^2}}}.


From this equation,  

    {{{r^2}}} = {{{15552/432}}} = 36.


Hence,  r = {{{sqrt(36)}}} = +/- 6.


Thus, there are TWO geometric progressions satisfying the given properties:
one progression with the common ratio r= 6 and the other progression with the common ratio r= -6.



For the first progression

    {{{a[12]}}} = {{{a[5]*r^(12-5)}}} = {{{15552*6^7}}} = 4353564672.



For the second progression

    {{{a[12]}}} = {{{a[5]*r^(12-5)}}} = {{{15552*(-6)^7}}} = -4353564672.



Thus there are two answers and two values for the 12-th term: 4353564672 and -4353564672.
</pre>

Solved.


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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.