Question 1202408
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The third term of a geometric progression is nine times the first term. 
The sum of the first six terms is k times the sum of the first two terms. Find the value of k.
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<pre>
{{{S[6]}}} = {{{a[1]}}} + {{{a[2]}}} + {{{a[3]}}} + {{{a[4]}}} + {{{a[5]}}} + {{{a[6]}}}.


Group the terms

{{{S[6]}}} = ({{{a[1] + a[2]}}}) + ({{{a[3] + a[4]}}}) + ({{{a[5] + a[6]}}}).



You are given  {{{a[3]}}} = {{{9*a[1]}}}.

It implies  {{{a[4]}}} = {{{9*a[2]}}};  {{{a[5]}}} = {{{81*a[1]}}};  {{{a[6]}}} = {{{81*a[2]}}}.


Therefore

{{{S[6]}}} = {{{(a[1] + a[2])}}} + {{{9*(a[1] + a[2])}}} + {{{81*(a[1] + a[2])}}} = {{{(1 + 9 + 81)*(a[1]+a[2])}}} = {{{91*(a[1]+a[2])}}}.


Thus the coefficient k is equal to 91.


<U>ANSWER</U>.  k = 91.
</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.


Learn the subject from there.