SOLUTION: The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive

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Question 988607: The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive
Found 3 solutions by MathLover1, KMST, MathTherapy:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
if the fourth term exceeds the third term by , we have
....eq.1
if the sum of the second and third term is , we have
....eq.2
By using we have:

....eq.1
....eq.2
---------------------------------------------------add eq.2 and eq.1




.............substitute in eq.2

....eq.2


......divide by





=> one solution will be: =>
=> another solution will be: =>=>
now we can find first term:
if
....eq.2




if :
....eq.2
......multiply by




so, there are two solutions:
1. and
2. and


then the second term is:
if and :


third term is


fourth term



the terms of sequence are:=>


if ,


.
third term is



and fourth term is



the terms of sequence are:=>

now, check given data:
for =>
....eq.1


....eq.2



for =>
....eq.1


....eq.2
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
= the common ratio.
= the second term.
= the third term.
= the fourth term.
The fourth term of a geometric progression exceeds the third term by 54 translates as
<--><--> .
The sum of the second and third term is 36 translates as
<--><--> .
The ratio both equations tells us that
-->-->-->}--> .
The equation is a quadratic equation.
As is true for all quadratic equations, it can be solved by "completing the square, or by using the quadratic formula.
This particular quadratic equation can also be solved by factoring:
--->--->}--->---> .
If the common ratio is positive, it must be .
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive
Since , then , , and 



Since  exceeds  by 54, then we can say that:  



 ---------- eq (i)



Since the sum of  and  is 36, then we can say that: 



 --------- eq (ii)



Since  and , we can then say that: 

 ------ Cross-multiplying

 -------- Factoring out GCF, 18











(r - 3)(2r + 1) = 0

Common ratio, or 		OR           (ignore, since r MUST be > 0) 

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