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October 2nd, 2019, 09:47 AM  #1 
Newbie Joined: Oct 2019 From: San diego Posts: 3 Thanks: 0  Geometric sequence
How can I find x so that a+x, b+x, c+x is a geometric sequence?
Last edited by skipjack; October 2nd, 2019 at 07:52 PM. 
October 2nd, 2019, 10:05 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,035 Thanks: 2271 
You need (a + x)(c + x) = (b + x)².

October 2nd, 2019, 10:17 AM  #3 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 
Now just set a,b,c the way you want and find their values. Note: If (p,q,r) is a geometric sequence, (r,q,p) is also a geometric sequence. Last edited by skipjack; October 2nd, 2019 at 07:51 PM. 
October 2nd, 2019, 04:37 PM  #4 
Newbie Joined: Oct 2019 From: San diego Posts: 3 Thanks: 0  Found a solution 
October 2nd, 2019, 04:38 PM  #5  
Newbie Joined: Oct 2019 From: San diego Posts: 3 Thanks: 0  Funny but true Quote:
1,3,9 and 9,3,1 are both geometric sequences. First one has a ratio of 3, the second one has a ratio of 1/3  
October 2nd, 2019, 07:54 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 21,035 Thanks: 2271 
What if no such x exists?

October 3rd, 2019, 02:54 AM  #7 
Senior Member Joined: Feb 2010 Posts: 714 Thanks: 151 
I think that will happen if $\displaystyle b$ is the arithmetic mean of $\displaystyle a$ and $\displaystyle c$.

October 3rd, 2019, 03:01 AM  #8  
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98  Quote:
Also {8,4,2} where the ratio is $\displaystyle q=1/2$.  

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