Geometric sequence

Oct 2019
3
0
San diego
How can I find x so that a+x, b+x, c+x is a geometric sequence?
 
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skipjack

Forum Staff
Dec 2006
21,482
2,472
You need (a + x)(c + x) = (b + x)².
 
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Dec 2015
1,085
169
Earth
Now just set a,b,c the way you want and find their values.:D

Note: If (p,q,r) is a geometric sequence, (r,q,p) is also a geometric sequence.
 
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Oct 2019
3
0
San diego
Funny but true

Now just set a,b,c the way you want and find their values .:D

Note: If {p,q,r} is a geometric sequence then {r,q,p} is also a geometric sequence .
You're actually correct

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
 

skipjack

Forum Staff
Dec 2006
21,482
2,472
What if no such x exists?
 
Feb 2010
739
162
I think that will happen if \(\displaystyle b\) is the arithmetic mean of \(\displaystyle a\) and \(\displaystyle c\).
 
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Dec 2015
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Earth
I think that will happen if \(\displaystyle b\) is the arithmetic mean of \(\displaystyle a\) and \(\displaystyle c\).
for x=1 : {1+a,1+b,1+c}. Now set a=1,b=3,c=7 and we have a geometric sequence {2,4,8} where the ratio is \(\displaystyle q=2\).
Also {8,4,2} where the ratio is \(\displaystyle q=1/2\).