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Find the 8th term of the arithmetic sequence 5x + 7, x +12, -3x + 17
In an arithmetic sequence, the difference between consecutive terms is always the same. To find the common difference, we can subtract any term from its previous term.
First, we'll find the common difference:
Common difference = (x+12) - (5x+7)
= x+12 - 5x - 7
= -4x + 5
We can check that the common difference is the same for the other pairs of consecutive terms.
To find the 8th term, we'll use the formula:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the term we want to find, and d is the common difference.
We already know that the common difference is -4x + 5.
So, to find the first term, we plug in n=1:
a1 = 5x + 7
Now we can find the 8th term:
a8 = a1 + 7d
= (5x + 7) + 7(-4x + 5)
= 5x + 7 - 28x + 35
= -23x + 42
Therefore, the 8th term of the given arithmetic sequence is -23x + 42.
Find the equations of the asymptotes of the hyperbola defined by the equation shown below. If necessary, round to the nearest tenth.
−64x^2+ 9y^2- 1024x + 144y-4096 = 0
Asymptotes: y=
and y=
To find the equations of the asymptotes of a hyperbola, we need to first put the given equation in standard form. The standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the x-axis is:
[(x - h)^2 / a^2] - [(y - k)^2 / b^2] = 1
where a and b are the lengths of the semi-major and semi-minor axes, respectively.
To put the given equation in standard form, we need to complete the square for both the x and y terms:
-64x^2 - 1024x + 9y^2 + 144y = 4096
-64(x^2 + 16x) + 9(y^2 + 16y) = 4096 + 64*16 - 9*16
-64(x + 8)^2 + 9(y + 8)^2 = 4225
[(x + 8)^2 / (-4225/64)] - [(y + 8)^2 / (4225/9)] = 1
Comparing this to the standard form, we see that:
h = -8
k = -8
a^2 = -4225/64
b^2 = 4225/9
Since the transverse axis is parallel to the x-axis, the asymptotes will have the form:
y = ± (b/a) (x - h) + k
Plugging in the values from the standard form, we get:
y = ± (√(4225/9) / √(-4225/64)) (x + 8) - 8
y = ± (64/27) (x + 8) - 8
Simplifying and rounding to the nearest tenth, we get:
y = (2.4)x - 31.6 and y = (-2.4)x + 15.6 as the equations of the asymptotes.
To find the 8th term of the arithmetic sequence, we need to determine the common difference first.
The difference between any two consecutive terms in an arithmetic sequence is constant.
Let's find the common difference (d) between the second and first terms:
(x + 12) - (5x + 7) = -3x + 17 - (x + 12)
Rearranging the terms:
-4x + 5 = -2x + 5
Combining like terms:
-4x + 2x = 5 - 5
-2x = 0
Dividing by -2:
x = 0
Now that we have found the value of x, we can substitute it into any of the terms given to find the first term (a₁) of the arithmetic sequence. Let's use the first term, 5x + 7:
a₁ = 5(0) + 7 = 7
So, the first term of the arithmetic sequence is 7.
Now, we can find the common difference (d) between any two consecutive terms:
d = (x + 12) - (5x + 7)
= 0 + 12 - 5(0) - 7
= 12 - 7
= 5
Therefore, the common difference is 5.
To find the 8th term, we can use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n - 1)d
Substituting the known values:
a₈ = 7 + (8 - 1)(5)
= 7 + 7(5)
= 7 + 35
= 42
Therefore, the 8th term of the arithmetic sequence is 42.