SOLUTION: If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for a) x b)the common ratio c)the sum of the Gp pls show workings

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Question 1210418: If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for
a) x
b)the common ratio
c)the sum of the Gp
pls show workings
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for
a) x
b)the common ratio
c)the sum of the Gp
pls show workings

Let the common ratio be r

Then
system%28%0D%0A%283-x%29%2Ar+=+%286%29%2C%0D%0A%286%29%2Ar+=+%287-5x%29%29

Solve the first equation for r:

%283-x%29%2Ar+=+%286%29
r+=+6%2F%283-x%29

Substitute the right side of that for r in the second equation:

%286%29%2Ar+=+%287-5x%29

%286%29%2A%286%2F%283-x%29%29+=+%287-5x%29

36%2F%283-x%29+=+7-5x
36=%283-x%29%287-5x%29
36=21-22x%2B5x%5E2
15%2B22x-5x%5E2=0
-5x%5E2%2B22x%2B15=0
5x%5E2-22x-15=0
%285x%2B3%29%28x-5%29=0
5x+3=0; x-5=0
5x=-3;    x=5 
x=-3/5

Substitute x=-3/5 in 

r+=+6%2F%283-x%29
r+=+6%5E%22%22%2F%283-%28-3%2F5%29%29
r+=+6%5E%22%22%2F%283%2B3%2F5%29
Multiply top and bottom by 5
r+=+30%2F%2815%2B3%29%29%29%0D%0A%7B%7B%7Br+=+30%2F18
r+=+5%2F3

Substitute x=5 in 

r+=+6%2F%283-x%29
r+=+6%2F%283-5%29
r+=+6%2F%28-2%29
r+=+-3

a)x = -3/5 or 5
b)the common ratio = r = 5/3 or r = -3
c)the sum of the GP: 
if x = -3/5               if x = 5
%283-x%29%2B%286%29%2B%287-5x%29        ditto 
3-x%2B6%2B7-5x            ditto
16-6x                   ditto
16-6%28-3%2F5%29             16-6%285%29 
16%2B18%2F5                 16-30
80%2F5%2B18%2F5                 -14
98%2F5 
Sum of GP = 98/5  or sum of GP = -14                 

Edwin

Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


Another tutor has shown one common method for finding the two possible values of x that produce a geometric sequence, and he has then proceeded to answer all the parts of the question.

Here is an alternative method for finding the two possible values of x and hence of the common ratio. (See his response for the remainder of the problem.)

A very useful method for solving many problems involving geometric sequences is using the fact that if A, B, and C are successive terms of the sequence, then

B%5E2=AC

So in this problem,

6%5E2=%283-x%29%287-5x%29
36=21-22x%2B5x%5E2
5x%5E2-22x-15=0
%285x%2B3%29%28x-5%29=0
5x%2B3=0 or x-5=0
x+=+-3%2F5 or x=5