1. The fourth and seventh terms of a geometric sequence are 6 and 384, respectively. What is the common ratio and the sixth term of the sequence?
2. The fifth and eight terms of a geometric sequence of real numbers are 7! and 8! respectively. What is the first term?
384/6 = 64
There are three terms, so r = 4
T6 = T4*r^2 = 6*16 = 96
r^3 = 8!/7! = 8
so, r=2
T1 = T5/r^4 = 7!/16 = 315
To find the common ratio and the sixth term of a geometric sequence:
1. Use the formula for the nth term of a geometric sequence to find the common ratio. The formula is: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term.
From the given information, we have the following two equations:
a4 = a1 * r^(4-1) = 6
a7 = a1 * r^(7-1) = 384
Divide these two equations to eliminate a1:
(a7 / a4) = (a1 * r^(7-1)) / (a1 * r^(4-1))
384 / 6 = r^6 / r^3
48 = r^6 / r^3
48 = r^(6-3)
48 = r^3
Take the cube root of both sides to solve for the common ratio:
r = ∛48
r ≈ 3.634
2. Now, use the formula for the nth term of a geometric sequence to find the sixth term. Substitute the known values into the formula:
a6 = a1 * r^(6-1)
a6 = a1 * r^5
Substitute the value of r:
a6 = a1 * (3.634)^5
Simplify the expression:
a6 ≈ a1 * 275.382
Therefore, the common ratio is approximately 3.634, and the sixth term of the sequence is approximately 275.382 * a1.
To find the first term of a geometric sequence:
1. Use the formula for the nth term of a geometric sequence to find the first term. The formula is: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term.
From the given information, we have the following two equations:
a5 = a1 * r^(5-1) = 7!
a8 = a1 * r^(8-1) = 8!
Divide these two equations to eliminate a1:
(a8 / a5) = (a1 * r^(8-1)) / (a1 * r^(5-1))
8! / 7! = r^7 / r^4
8 = r^7 / r^4
Take the fourth root of both sides to solve for the common ratio:
r = ∜8
r = 2
2. Now, use the formula for the nth term of a geometric sequence to find the first term. Substitute the known values into the formula:
a5 = a1 * r^(5-1)
7! = a1 * 2^4
Simplify the expression:
7! = 16 * a1
To solve for a1, divide both sides by 16:
a1 = 7! / 16
Therefore, the first term of the geometric sequence is 7! / 16.
To find the common ratio and the sixth term of a geometric sequence, we can use the formula for the nth term of a geometric sequence. The formula is given by:
a_n = a * r^(n-1)
where a_n is the nth term, a is the first term, r is the common ratio, and n is the position of the term.
Let's solve the first problem:
1. The fourth term is given as 6, which means a_4 = 6.
2. The seventh term is given as 384, which means a_7 = 384.
From the formula, we can set up two equations using the given information:
a_4 = a * r^(4-1) ... (1)
a_7 = a * r^(7-1) ... (2)
Substituting a_4 = 6 and a_7 = 384 into equations (1) and (2), we get:
6 = a * r^3 ... (3)
384 = a * r^6 ... (4)
To find the common ratio, we can divide equation (4) by equation (3):
384/6 = (a * r^6)/(a * r^3)
64 = r^3
Take the cube root of both sides:
∛64 = r
Simplifying the cube root of 64, we get r = 4.
Now that we have the common ratio, we can find the sixth term (a_6). Substitute the common ratio (r = 4) into equation (3):
6 = a * (4^3)
6 = a * 64
a = 6/64
a = 3/32
Therefore, the common ratio is 4, and the sixth term of the sequence is 3/32.
Let's solve the second problem:
2. The fifth term is given as 7!, which means a_5 = 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1. Similarly, the eighth term is given as 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, which means a_8 = 8!.
Again, we can set up two equations using the given information:
a_5 = a * r^(5-1) ... (5)
a_8 = a * r^(8-1) ... (6)
Substituting the values into equations (5) and (6), we get:
7! = a * r^4 ... (7)
8! = a * r^7 ... (8)
To find the first term (a), we can divide equation (8) by equation (7):
8! / 7! = (a * r^7) / (a * r^4)
8 = r^3
Take the cube root of both sides:
∛8 = r
Simplifying the cube root of 8, we get r = 2.
Now that we have the common ratio, we can find the first term (a) by substituting the common ratio (r = 2) into equation (7):
7! = a * (2^4)
7! = a * 16
a = (7!) / 16
Therefore, the first term of the sequence is (7!) / 16.