In a geometric sequence, the fourth term is -4, and the eighth term is -324. What is the tenth term?
t(4) = ar^3 = -4
t(8) = ar^7 = -324
divide t(8) by t(4)
r^4 = 81
r = 81^(1/4) = ± 3
if r = +3
in t(4)
a(3^3) = -4
a = -4/27
t(10) = (-4/27)(3^9) = -2916
if r = -3
a = -4/-27 = 4/27
t(10) = ar^9 = (4/27)(-3)^9 = 4(-729) = -2916
either way, t(10) = -2916
Why did the geometric sequence go to the therapist? Because it couldn't find its common ratio!
But let me use my clown math skills to help you out. Let's call the first term "a" and the common ratio "r". We know that the fourth term is -4, so we have:
a * r^3 = -4
And we also know that the eighth term is -324, so:
a * r^7 = -324
Now, let's divide these two equations to get rid of "a":
(r^7)/(r^3) = (-324)/(-4)
Simplifying the left side gives us:
r^(7-3) = 81
So, we have r^4 = 81. Taking the fourth root of both sides gives us r = 3.
Now, we can plug this value of r back into either of the previous equations to find "a". Let's use the first equation:
a * 3^3 = -4
a * 27 = -4
Dividing both sides by 27, we get:
a = -4/27
Finally, we can find the tenth term using the formula:
n = a * r^(n-1)
n = (-4/27) * 3^(10-1)
n = (-4/27) * 3^9
n ≈ -4 * 19683/27
Therefore, the tenth term is approximately -1456.
To find the tenth term of the geometric sequence, we need to determine the common ratio first.
Let's denote the first term as "a" and the common ratio as "r".
We are given that the fourth term is -4, so we can write the following equation for the fourth term:
a * r^3 = -4 (1)
We are also given that the eighth term is -324, so we can write the following equation for the eighth term:
a * r^7 = -324 (2)
Now, we can divide equation (2) by equation (1) to eliminate the first term "a":
(a * r^7) / (a * r^3) = -324 / -4
Simplifying this equation gives:
r^4 = 81
Taking the fourth root of both sides, we find that the common ratio "r" is:
r = ±3
Since this is a geometric sequence, we should choose only one value for the common ratio.
Now, we have two cases to consider:
Case 1: r = 3
Substituting this value into equation (1):
a * 3^3 = -4
27a = -4
a = -4 / 27
So, in this case, the first term "a" is approximately -0.148.
Now, we can find the tenth term using the formula:
T_n = a * r^(n - 1)
T_10 = (-0.148) * 3^(10 - 1)
T_10 = (-0.148) * 3^9
T_10 = (-0.148) * 19683
T_10 ≈ -2911.404
Case 2: r = -3
Substituting this value into equation (1):
a * (-3)^3 = -4
-27a = -4
a = -4 / -27
a ≈ 0.148
Following the same steps as in case 1, we can find the tenth term:
T_10 = (0.148) * (-3)^(10 - 1)
T_10 = (0.148) * (-3)^9
T_10 = (0.148) * -19683
T_10 ≈ -2911.404
Therefore, the tenth term of the geometric sequence is approximately -2911.404, regardless of the value of the common ratio.
To find the tenth term in a geometric sequence, we need to determine the common ratio of the sequence and use it to calculate the value.
In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio (r).
Let's denote the first term as 'a' and the common ratio as 'r'.
Given that the fourth term is -4 and the eighth term is -324, we can form the following equations:
a * r^3 = -4 -- Equation 1
a * r^7 = -324 -- Equation 2
To solve these equations, we can divide Equation 2 by Equation 1 to eliminate 'a' and get an expression for 'r':
(r^7) / (r^3) = (-324) / (-4)
r^4 = 81
r = ∛(81)
r = 3
Now that we have determined the common ratio 'r' as 3, we can use it to find the tenth term. However, we first need to determine the first term 'a'.
Using Equation 1:
a * (3^3) = -4
a * 27 = -4
a = -4 / 27
Now that we know the value of 'a' as -4/27 and the common ratio 'r' as 3, we can find the tenth term using the following formula:
Tn = a * r^(n-1)
Substituting the values:
T10 = (-4/27) * (3^(10-1))
T10 = (-4/27) * (3^9)
T10 ≈ -1968.74
Therefore, the tenth term of the geometric sequence is approximately -1968.74.