Given that 6,x, -4,and y are consecutive terms in an arithmetic progression, find the value of (xy)
To find the value of (xy), we need to determine the values of x and y first.
Given that 6, x, -4, and y are consecutive terms in an arithmetic progression, we can use the formula for arithmetic progression to find the common difference (d) between the terms.
The formula for the nth term of an arithmetic progression is given by:
an = a1 + (n-1)d
where:
an is the nth term
a1 is the first term
n is the position of the term
d is the common difference
Let's use this formula to find the value of d.
The second term is x, which is the same as a2.
Since a2 = a1 + (2-1)d, we can substitute the known values:
x = 6 + d
Similarly, the fourth term is y, which is the same as a4.
Since a4 = a1 + (4-1)d, we can substitute the known values:
y = 6 + 3d
Now we have two equations:
x = 6 + d Equation 1
y = 6 + 3d Equation 2
To solve this system of equations, we need to eliminate the variable d. We can do this by subtracting Equation 1 from Equation 2:
y - x = 6 + 3d - (6 + d)
y - x = 3d + 3d
y - x = 6d
Now we have an expression that relates y, x, and d: y - x = 6d.
Since we know that x and y are consecutive terms, the difference between them is the common difference, d.
Now let's substitute the known values to find the value of d:
-4 - 6 = 6d
-10 = 6d
Divide both sides of the equation by 6:
d = -10/6
d = -5/3
Now that we have the value of the common difference d, we can substitute it back into one of the equations to find the values of x and y.
Let's use Equation 1, which is x = 6 + d:
x = 6 + (-5/3)
x = 6 - (5/3)
x = 18/3 - 5/3
x = 13/3
Now we have the values of x and d:
x = 13/3
d = -5/3
To find the value of (xy), we can substitute these values into the expression:
xy = (13/3) * y
Since y = 6 + 3d, we can substitute the value of d into this expression to get the value of y:
y = 6 + 3(-5/3)
y = 6 - 5
y = 1
Now we have the values of x and y:
x = 13/3
y = 1
Substituting these values into the expression for (xy):
xy = (13/3) * 1
xy = 13/3
Therefore, the value of (xy) is 13/3.
To find the value of (xy), we need to determine the value of x and y in the given arithmetic progression.
First, let's find the common difference (d) between the consecutive terms. In an arithmetic progression, the difference between any two consecutive terms is constant.
From the given information, we have the following terms in the arithmetic progression: 6, x, -4, y.
To find the common difference, we can use the formula for the nth term in an arithmetic progression:
Tₙ = a + (n - 1) * d
where Tₙ represents the nth term, a is the first term, n is the position of the term, and d is the common difference.
For the first term (a) and the third term (T₃), we have:
a = 6
T₃ = -4
Substituting these values into the formula, we get:
-4 = 6 + (3 - 1) * d
-4 = 6 + 2d
-10 = 2d
d = -5
Now that we have the common difference (d), we can find the values of x and y.
To find x, we need to determine the position of x in the arithmetic progression. It is the second term (T₂).
Using the formula again, we have:
T₂ = 6 + (2 - 1) * (-5)
T₂ = 6 - 5
T₂ = 1
So, x = 1.
To find y, we need to determine the position of y in the arithmetic progression. It is the fourth term (T₄).
Using the formula, we have:
T₄ = 6 + (4 - 1) * (-5)
T₄ = 6 + 3 * (-5)
T₄ = 6 - 15
T₄ = -9
So, y = -9.
Finally, to find the value of (xy):
(xy) = (1 * -9)
(xy) = -9
Therefore, the value of (xy) is -9.
6 + a = x
6 + 2 a = -4
12 + 2 a = 2 x
6 + 2 a = -4
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6 = 2 x + 4
2 x = 4
x = 2
a = 2 - 6 = -4
so
6, 2 , -4 , -8
2 * -8 = -16