X-8;x;2x-5 are the first terms of an arithmetic
1. Determine the values of x
2. Determine the general term
3. Determine the values of the 115 term
x+1;2x;5x+5;... is an arithmetic sequence
1. Determine the values of x
2. Determine the general term
2x-1;x-3;1-3x are the first three terms of an arithmetic sequence
1. Determine the values of x
2. Determine the general term
1. To determine the values of x for the first sequence (X-8, x, 2x-5), we need to find the common difference between the terms.
The common difference d is found by subtracting the previous term from the current term. So,
(x) - (X-8) = (2x-5) - (x)
Simplifying the equation, we get:
x - X + 8 = 2x - 5 - x
8 - 5 = 2x - x - X
3 = x - X
Since we have two variables, we need another equation to solve for the values of x. We can use the second term x:
x = 2x - 5
Solving for x, we find:
x - 2x = -5
-1x = -5
x = 5
Now, substituting the value of x into the first equation (3 = x - X), we find:
3 = 5 - X
-2 = -X
X = 2
Therefore, the values of x for the first sequence are x = 5 and X = 2.
2. To determine the general term, we need to find the common difference first.
The common difference d is found by subtracting the previous term from the current term.
The common difference between the second and first term is:
x - (X-8) = x - X + 8 = 3
Therefore, the general term can be written as:
Tn = a + (n-1)d
Tn = X + (n-1)(x-X+8)
Tn = 2 + (n-1)(5-2+8)
Tn = 2 + (n-1)(11)
Tn = 2 + 11n - 11
Tn = 11n - 9
3. To determine the value of the 115th term, we substitute n = 115 into the general term equation:
T115 = 11(115) - 9
T115 = 1265 - 9
T115 = 1256
Therefore, the value of the 115th term is 1256.
For the second sequence (x+1, 2x, 5x+5):
1. To determine the values of x, we need to find the common difference between the terms.
The common difference d is found by subtracting the previous term from the current term. So,
(2x) - (x+1) = (5x+5) - (2x)
Simplifying the equation, we get:
2x - x - 1 = 5x - 2x + 5
x - 1 = 3x + 5
Bringing like terms to one side, we have:
x - 3x = 5 + 1
-2x = 6
x = -3
Therefore, the value of x for the second sequence is x = -3.
2. To determine the general term, we need to find the common difference first.
The common difference between the second and first term is:
(2x) - (x+1) = 3x + 3 - x - 1 = 2x + 2
Therefore, the general term can be written as:
Tn = a + (n-1)d
Tn = (x+1) + (n-1)(2x+2)
Tn = (-3+1) + (n-1)(2*-3+2)
Tn = -2 + (n-1)(-4)
Tn = -2 - 4n + 4
Tn = 2 - 4n
Therefore, the general term for the second sequence is Tn = 2 - 4n.
For the third sequence (2x-1, x-3, 1-3x):
1. To determine the values of x, we need to find the common difference between the terms.
The common difference d is found by subtracting the previous term from the current term. So,
(x-3) - (2x-1) = (1-3x) - (x-3)
Simplifying the equation, we get:
x - 3 - 2x + 1 = 1 - 3x - x + 3
-x - 2 = 4 - 4x
Bringing like terms to one side, we have:
x + 4x = 4 + 2
5x = 6
x = 6/5
Therefore, the value of x for the third sequence is x = 6/5.
2. To determine the general term, we need to find the common difference first.
The common difference between the second and first term is:
(x-3) - (2x-1) = x - 3 - 2x + 1 = -x -2
Therefore, the general term can be written as:
Tn = a + (n-1)d
Tn = (2x-1) + (n-1)(-x-2)
Tn = (2*(6/5) - 1) + (n-1)(-(6/5) - 2)
Tn = (12/5 - 1) + (n-1)(-(16/5))
Tn = (7/5) + (n-1)(-16/5)
Tn = 7/5 - 16(n-1)/5
Tn = (7-16n+16)/5
Tn = (23-16n)/5
Therefore, the general term for the third sequence is Tn = (23-16n)/5.