Question 1140428
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<pre>
The formula for the sum of the first "n" terms of any geometric progression


    {{{S[n]}}} = {{{(a[1]*q^n-a[1])/(q-1)}}}


(where "q" is the common ratio) can be written in an equivalent form 


    {{{S[n]}}} = {{{(a[n]*q - a[1])/(q-1)}}}.


So, with the given data,


    1533 = {{{(768*q-3)/(q-1)}}},


or, simplifying


    768*q - 3 = 1533*(q-1)

    768q - 3 = 1533q - 1533

    1533 - 3 = 1533q - 768q

    1530 = 765q

    q = {{{1530/765}}} = 2.


So, the common ratio is just found: it is 2.


Next,  to find "n", the number of terms, use the general formula for the n-th term


    768 = {{{3*2^(n-1)}}}

    {{{2^(n-1)}}} = {{{768/3}}} = 256

    ============>  n - 1 = 8;  hence,  n = 9.


<U>ANSWER</U>.  The number of terms is 9 and the common ratio is 2.
</pre>

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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.