SOLUTION: Consider the sequence x-3, x+1, 2x+8. One value for x is 5, making the sequence geometric. find the other value of x for which the sequence is geometric For this value of x find

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Question 529087: Consider the sequence x-3, x+1, 2x+8. One value for x is 5, making the sequence geometric.
find the other value of x for which the sequence is geometric
For this value of x find the common ratio and the sum of the infinite sequence
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
For the sequence to be geometric the ratio between consecutive terms must be the same.

You could use that definition to find your .
Or you could use the fact that in a geometric sequence each middle term is the geometic mean of the neighboring terms. If a, b, and c are consecutive terms in a geometric sequence that means
or
Either way, you end up with

which simplifies to

so the solutions are
and
That makes the first three terms -8, -4, and -2
and you should get the ratio and sum from that easily.
Without even using the formula for sum of a geometric sequence I realize that
-8+(-4)+(-2)+(-1)+(-1/2)+ ... gets closer and closer to -16, and the difference is always equal to that shrinking last term.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
The other tutor's solution is incorrect.

x-3, x+1, 2x+8

Let r = the common ratio 

Then we have the system of two equations in two unknowns:

r(x-3) = x+1
r(x+1) = 2x+8

Solving each for r:

r = 
r = 

Setting the right sides equal to each other, since both
equal to r:

 = 

Cross-multiplying:

(x+1)(x+1) = (x-3)(2x+8)

x² + 2x + 1 = 2x² + 2x - 24

0 = x² - 25

0 = (x - 5)(x + 5)

 x - 5 = 0   x + 5 = 0
     x = 5       x = -5

As they told us, x = 5 is one of the values and it
makes the sequence

x-3, x+1, 2x+8 become

5-3, 5+1, 2(5)+8

2, 6, 18

and the common ratio is  =  = 3.

The other value of x is -5.  It makes the sequence

x-3, x+1, 2x+8 become:

-5-3, -5+1, 2(-5)+8

-8, -6, -2

and the ratio is  =  =   

Since the common ratio is less than 1, we can sum the series
to infinity with the equation:

 = 

where  is the first term -8, and r = 

Substituting:

 =  =  =  = -12.

Edwin
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