if (3-x), (6),(7-x) is a geometric sequence calculate (a)the two possible values of x (b) common ratio (c) sum of the sequence
since there is a common ratio,
6/(3-x) = (7-x)/6
so, solve for x
use that to find the ratio.
Then you can find the 1st term and use the GP sum formula.
I must say, not the ratio I expected...
To determine the values of x in the geometric sequence (3 - x), 6, (7 - x), we can use the definition of a geometric sequence and the fact that consecutive terms in a geometric sequence have a common ratio.
(a) Finding the values of x:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor (the common ratio). So, to find the values of x, we need to set up an equation based on the given sequence:
6 / (3 - x) = (7 - x) / 6
Next, we cross-multiply and simplify the equation:
6 * (7 - x) = (3 - x) * 6
42 - 6x = 18 - 6x
Notice that the values on both sides cancel each other out. This means that any value of x will satisfy this equation, resulting in infinitely many solutions. Thus, there are an infinite number of possible values for x.
(b) Finding the common ratio:
To find the common ratio (r) of a geometric sequence, we divide any term in the sequence by its preceding term. In this case, we can choose any two consecutive terms:
Common ratio (r) = (7 - x) / 6 = 6 / (3 - x)
Cross-multiplying and simplifying, we get:
6 * (7 - x) = 6 * (3 - x)
42 - 6x = 18 - 6x
As we can see, the equation simplifies to 42 - 6x = 18 - 6x. The variable x cancels out, indicating that the common ratio is not dependent on x. Therefore, the common ratio is 6/3 = 2.
(c) Finding the sum of the sequence:
The formula for calculating the sum (S) of a geometric sequence with n terms, starting with the first term a and common ratio r, is given by:
S = a * (1 - r^n) / (1 - r)
In our case, the first term a is (3 - x), the common ratio r is 2, and we have three terms.
S = (3 - x) * (1 - 2^3) / (1 - 2)
Simplifying further:
S = (3 - x) * (1 - 8) / (1 - 2)
S = (3 - x) * -7 / -1
S = 7(3 - x)
Therefore, the sum of the sequence is 7(3 - x).