Question 1068007
.
The three consecutive term of an exponential sequence  are the second, third and sixth term of a linear sequence, find the common ratio of g.p 
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This formulation is . . . mmm . . . how to say it . . . is FAR FROM TO BE PERFECT.


Let me re-formulate it to make it mathematically correct:


<pre>
     The three consecutive terms of a geometric sequence  are the second, third and sixth term of an arithmetic sequence. 
     Find the common ratio of the GP.
</pre>

Now it is easy to solve it.

<pre>
1.  Our numbers (1-st, 2nd and 3-rd) are a, ar and ar^2, three consecutive terms of an geometric progression 
    with the first term "a" and the common ratio "r".


2.  Since a and ar are the consecutive terms of an arithmetic progression, their difference is "d", the common difference of the AP

    ar - r = d.       (1)


    Since ar and ar^2 are the second and the sixth terms of the AP, their difference is four times d": 

    {{{ar^2 - ar}}} = 4d.     (2)


3.  From (1) and (2) you have this equation

    4(ar-a) = {{{ar^2 - ar}}},   which is equivalent to

    4a*(r-1) = ar*(r-1).


    Assuming that  a=/=0 and r=/=1, we can cancel the factors "a" and (r-1) in both sides and to get

    4 = r.


    It is our answer: r = 4.


4.  Now let us consider these exclusive cases  a = 0  and  r= 1.


    a) if a = 0, then the three terms of the GP are 0, 0, and 0.

       This progression satisfies all the problem conditions, but it is not very interesting. It is a degenerated case. 
       But still satisfies all conditions.


    b) if r = 1, then the three terms of the GP are a, a, and a,  for any arbitrary "a". Considered as the arithmetic progression, 
       it has the common difference 0 (zero). 

       Its 6-th term is also "a", and such a set satisfies all the problem conditions, again. Although is degenerated, too.


5.  Therefore, the answer is: There is a unique non-degenerated solution with the common ratio 4.
                              There are infinitely many degenerated solutions with the common ratio 1.
</pre>

Solved.


On arithmetic progression see the lessons

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

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

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

in this site.


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>

in this site.



Also, 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 topics 
<U>"Arithmetic progressions"</U> and <U>"Geometric progressions"</U>.