6x+12,2x+4 and x-7 are the first three terms of a geometric sequence.
-is this a converging or diverging sequence...justify your answer
If there is a common ratio, then you need
(2x+4)/(6x+12) = (x-7)/(2x+4)
x = 25
So, the common ratio is 54/162 = 1/3
Since |r| < 1, the sequence converges.
To determine if the sequence is converging or diverging, we need to check if the common ratio is between -1 and 1 (converging) or not (diverging).
We can find the common ratio by dividing each term by the previous term. Let's calculate it:
Common ratio for the second term:
(2x + 4) / (6x + 12) = 2/6 = 1/3
Common ratio for the third term:
(x - 7) / (2x + 4) = 1/3
Since the common ratio is 1/3, which is between -1 and 1, we can conclude that the sequence is a converging sequence.
To determine whether the given sequence is converging or diverging, we need to check if the common ratio (r) between consecutive terms is between -1 and 1.
Let's find the common ratio (r) of the geometric sequence using the given terms:
The first term is 6x + 12.
The second term is 2x + 4.
The common ratio (r) can be found by dividing the second term by the first term:
(r) = (2x + 4) / (6x + 12)
We can simplify this by factoring out a 2 from both terms:
(r) = 2(x + 2) / 2(3x + 6)
Canceling out the common factors of 2, we get:
(r) = (x + 2) / (3x + 6)
Next, we need to check if the common ratio (r) lies between -1 and 1. We can do this by setting up an inequality:
-1 < (x + 2) / (3x + 6) < 1
Now, let's solve this inequality to determine the range of x values where the sequence would be converging:
-1 < (x + 2) / (3x + 6) < 1
Multiplying each term by (3x + 6) (which is positive):
-1(3x + 6) < x + 2 < 1(3x + 6)
-3x - 6 < x + 2 < 3x + 6
Simplifying, we get:
-6 < 2x + 2 < 6
Subtracting 2 from each term:
-8 < 2x < 4
Dividing each term by 2:
-4 < x < 2
Therefore, the sequence will be converging for values of x between -4 and 2.
To summarize, the sequence is converging for x values between -4 and 2 because the common ratio (r) lies between -1 and 1.