# Geometric sequence

#### JohnSecada2019

How can I find x so that a+x, b+x, c+x is a geometric sequence?

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#### skipjack

Forum Staff
You need (a + x)(c + x) = (b + x)Â².

1 person

#### idontknow

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.

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#### JohnSecada2019

Funny but true

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

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
What if no such x exists?

#### mrtwhs

I think that will happen if $$\displaystyle b$$ is the arithmetic mean of $$\displaystyle a$$ and $$\displaystyle c$$.

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#### idontknow

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 $$\displaystyle q=1/2$$.

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