Question 1083161
.
<pre>
Let "a" be the first term of the progression and "r" be the common ratio.

Then  {{{a[5]}}} = {{{a*r^4}}}  and  {{{a[7]}}} = {{{a*r^6}}} .

Therefore,  {{{a[7]/a[5]}}} = {{{r^2}}} = 2,  and,  hence,  {{{r}}} = {{{sqrt(2)}}}.


Then the sum is

254 = {{{a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6}}} = 


    = {{{a*(1 + sqrt(2) + (sqrt(2))^2 + (sqrt(2))^3 + (sqrt(2))^4 + (sqrt(2))^5 + (sqrt(2))^6)}}} = 


    = {{{a*(1 + sqrt(2) + 2 + 2*sqrt(2) + 4 + 4*sqrt(2) + 8)}}} = 


    = {{{a*(15 + 7*sqrt(2))}}}.


Hence,  a = {{{254/(15 + 7*sqrt(2))}}}.


<U>Answer</U>.  The first term is  a = {{{254/(15 + 7*sqrt(2))}}}.
</pre>

Solved.


On geometric progressions, see the 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>

&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>



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>.