Is this a geometric sequence?
x + 1, 2x + 2, 5x + 1 .....
check if
(2x+2)/(x+1) = (5x+1)/(2x+2) for ALL values of x
that is, the equation should reduce to 0 = 0 to be a GS
(I know it works for x = 3)
To determine if a sequence is geometric, we need to check if there is a common ratio between consecutive terms. Let's calculate the ratio between the second and first terms:
Ratio = (2x + 2) / (x + 1)
Next, let's calculate the ratio between the third and second terms:
Ratio = (5x + 1) / (2x + 2)
If the ratios are the same for all consecutive terms, then the sequence is geometric. Let's simplify the ratios and see if they are equal:
(2x + 2) / (x + 1) = (5x + 1) / (2x + 2)
Cross-multiplying:
(2x + 2)(2x + 2) = (x + 1)(5x + 1)
Expanding and rearranging the equation:
4x^2 + 8x + 4 = 5x^2 + 5x + x + 1
Simplifying further:
4x^2 + 8x + 4 = 5x^2 + 6x + 1
Subtracting both sides by (4x^2 + 6x + 1):
8x - 6x + 4 - 1 = 5x^2 - 4x^2
2x + 3 = x^2
Rearranging the equation:
x^2 - 2x - 3 = 0
Now, let's solve this quadratic equation either by factorization or using the quadratic formula. If we find distinct real solutions for x, then the sequence is not geometric. If we find equal real solutions for x, then the sequence is a degenerate geometric sequence.
Applying the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -2, and c = -3:
x = (2 ± √((-2)^2 - 4(1)(-3))) / 2(1)
x = (2 ± √(4 + 12)) / 2
x = (2 ± √16) / 2
x = (2 ± 4) / 2
x = (2 + 4) / 2 or x = (2 - 4) / 2
x = 6 / 2 or x = -2 / 2
x = 3 or x = -1
Since we have distinct real solutions for x, this means that the given sequence is not a geometric sequence.
To determine if a sequence is a geometric sequence, we need to check if the ratio between consecutive terms remains constant. Let's calculate the ratio between each consecutive term:
The ratio between the second and first term is (2x + 2)/(x + 1).
The ratio between the third and second term is (5x + 1)/(2x + 2).
If this is a geometric sequence, then both ratios should be equal. So, let's set up an equation:
(2x + 2)/(x + 1) = (5x + 1)/(2x + 2)
Now, we can solve this equation to check if it holds for all possible values of x.