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 GO.
6 = r * (3-x)
7-5x = r (6)
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r = 6/(3-x)
6 r + 5 x = 7
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[36 / (3-x)] + 5 x = 7
36 + 15x - 5 x^2 = 21 - 7 x
5 x^2 - 22 x - 15 = 0
(5x + 3)(x - 5) = 0
x = 5 or x = -3/5
you finish it
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To determine if the given polynomial expression is a geometric series, we need to check if the ratios between consecutive terms are constant.
The given expression is: (3-x) + 6 + (7-5x)
To find the ratios between consecutive terms, we derive each term.
First term: (3 - x)
Second term: 6
Third term: (7 - 5x)
Now, we can calculate the ratios:
Second term / First term = 6 / (3 - x) = 6 / (3 - x)
Third term / Second term = (7 - 5x) / 6
For this to be a geometric series, the ratios between consecutive terms must be equal. So, we can set up the following equations:
6 / (3 - x) = (7 - 5x) / 6
To solve for x, we can cross-multiply:
6 * (7 - 5x) = 6 * (3 - x)
42 - 30x = 18 - 6x
To simplify, we combine like terms:
30x - 6x = 42 - 18
24x = 24
Dividing both sides by 24:
x = 1
Now that we have found the value of x, we can substitute it back into the expression to find the common ratio and the sum of the geometric series.
The expression becomes: (3 - 1) + 6 + (7 - 5 * 1) = 2 + 6 + 2 = 10
Now we need to find the common ratio.
Using the first term (3 - x) and second term (6):
(3 - x) / 6 = (3 - 1) / 6 = 2 / 6 = 1/3
Hence, the common ratio is 1/3.
The sum of a geometric series can be calculated using the formula:
Sum of a Geometric Series = a * (1 - r^n) / (1 - r)
Where:
- a is the first term
- r is the common ratio
- n is the number of terms
In this case, we have only three terms, so n = 3.
Substituting the values into the formula:
Sum of the Geometric Series = (3 - 1) * (1 - (1/3)^3) / (1 - 1/3)
Simplifying:
Sum of the Geometric Series = 2 * (1 - 1/27) / (2/3)
Calculating:
Sum of the Geometric Series = 2 * (26/27) / (2/3)
Simplifying further:
Sum of the Geometric Series = 2 * (26/27) * (3/2)
Canceling out common factors:
Sum of the Geometric Series = 26/9
Therefore, the sum of the given geometric series is 26/9.
In summary:
a) The value of x is 1.
b) The common ratio is 1/3.
c) The sum of the geometric series is 26/9.