Question 1100391
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<pre>
The sum of an infinite GP with the first term "a" and the common difference "r", |r| < 1 is  {{{a/(1-r)}}}.


Therefore, our first equation for the given GP is

{{{a/(1-r)}}} = 16.        (1)


The second GP, comprised of the squares of the first GP, has the first term {{{a^2}}} and the common difference {{{r^2}}}.

Therefore, our second equation for the sum of squares is

{{{a^2/(1-r^2)}}} = {{{768/5}}}.      (2)


Now divide eq(2) by eq(1) (both sides). You will get

{{{a/(1+r)}}} = {{{48/5}}}.        (3)      (Take into account that {{{1-r^2}}} = (1-r)*(1+r))


Next step divide eq(3) by eq(1)  (both sides).  You will get

{{{(1-r)/(1+r)}}} = {{{3/5}}}.         (4)


Now we are at the finish line.  From eq(4), making cross-multiplying, you get

5*(1-r) = 3*(1+r)  ====>  5 - 5r = 3 + 3r  ====>  5-3 = 3r + 5r  ====>  8r = 2  ====>  r = {{{1/2}}}.


Thus we just found the common difference of the original progression.  It is  {{{1/2}}}.


Last step is to find "a":  a = 16*(1-r) = {{{16*(1-1/2)}}} = 8.


<U>Answer</U>.  The common ratio is {{{1/2}}},  the first term is 8,  and the 4-th term is  {{{8*(1/2)^3}}} = 1.
</pre>

Solved.


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There is a bunch of lessons on geometric progressions in this site

&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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-geometric-progressions.lesson>One characteristic property of geometric progressions</A>

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Fresh-sweet-and-crispy-problem-on-arithmetic-and-geometric-progressions.lesson>Fresh, sweet and crispy problem on arithmetic and geometric progressions</A>

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>



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.



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Thank you, Edwin.