If (x+1)1/2x and (x-5) are the 1st three term of a linear sequence find their common different.
I will assume your terms are
x+1, 1/2 x, and x-5
for a linear sequence:
(1/2)x - (x+1) = x-5 - (1/2)x
multiply each term by 2
x - 2(x+1) = 2x - 10 - x
x - 2x - 2 = 2x - 10 - x
-2x = -8
x = 4
so the terms are: 5, 2, and -1 which would give a common
difference of -3
Well, usually I'm more into clowning around than solving math problems, but I'll give it a shot! Let's break it down.
The first term of the sequence is x+1, the second term is 1/2x, and the third term is x-5.
To find the common difference, we need to see how much the terms increase or decrease by each time. Let's subtract the second term from the first term: (x+1) - (1/2x) = (2x + 2 - x)/2x = (x + 2)/2x.
Now let's subtract the third term from the second term: (1/2x) - (x-5) = (5 - (2x-1))/2x = (6 - 2x)/2x.
So, we've got two expressions: (x + 2)/2x and (6 - 2x)/2x, which represent the differences between consecutive terms.
Since these two expressions represent the same common difference, they must be equal to each other: (x + 2)/2x = (6 - 2x)/2x.
We can cancel out the 2x from both sides of the equation, leaving us with: x + 2 = 6 - 2x.
Now we can solve this simple equation: x + 2x = 6 - 2.
Combining like terms, we get: 3x = 4.
Dividing both sides by 3, we find: x = 4/3.
So, the common difference of the linear sequence is x = 4/3. Ta-da!
To find the common difference of a linear sequence, we need to look for a pattern between consecutive terms. Let's expand the expressions and observe:
(x + 1)^(1/2)x = x^(3/2) + x^(1/2)
(x - 5)
Next, let's write out the first three terms of the sequence:
Term 1: x^(3/2) + x^(1/2)
Term 2: (x^(3/2) + x^(1/2)) + (x - 5) = x^(3/2) + x^(1/2) + x - 5
Term 3: (x^(3/2) + x^(1/2) + x - 5) + (x - 5) = x^(3/2) + x^(1/2) + 2x - 10
To find the common difference, we'll subtract the second term from the first term, and then subtract the third term from the second term:
Term 2 - Term 1 = (x^(3/2) + x^(1/2) + x - 5) - (x^(3/2) + x^(1/2)) = x
Term 3 - Term 2 = (x^(3/2) + x^(1/2) + 2x - 10) - (x^(3/2) + x^(1/2) + x - 5) = x - 5
We have two different values when subtracting the terms, which indicates that there is no common difference. Therefore, the sequence given by the expressions does not form a linear sequence.
To find the common difference of a linear sequence, we need to examine the difference between consecutive terms.
Given the first three terms of the sequence: (x+1)1/2x, (x-5), and the term following (x-5), we can determine the common difference by subtracting consecutive terms.
The terms of the sequence can be written as:
1st term: (x+1)1/2x
2nd term: (x-5)
3rd term: (x-5) + d (where d is the common difference)
Now, subtracting consecutive terms:
2nd term - 1st term = (x-5) - (x+1)1/2x
Simplifying, we get: -1/2x - 6
3rd term - 2nd term = [(x-5) + d] - (x-5)
Simplifying, we get: d
Since the common difference can be found by subtracting consecutive terms, we set the above two expressions equal to each other and solve for d:
-1/2x - 6 = d
Hence, the common difference of the linear sequence is -1/2x - 6.