Find the common ratio r and the value of A1 with the information given.
1. A3=324 A7=64
2. A4=32/3 A8=54
A7/A3 = r^4 = 16/81, so r = 2/3
A8/A4 = r^4 = 54*3/32 = 81/16 = 3/2
To find the common ratio, we can use the formula for the nth term of a geometric sequence:
An = A1 * r^(n-1)
Let's solve each problem step-by-step:
1. For the first problem:
Given:
A3 = 324
A7 = 64
Using the formula, we can set up two equations:
A3 = A1 * r^(3-1) ---> 324 = A1 * r^2 ------(1)
A7 = A1 * r^(7-1) ---> 64 = A1 * r^6 ------(2)
Now, we can solve these equations to find the common ratio (r) and the value of A1:
Divide equation (2) by equation (1):
(64) / (324) = (A1 * r^6) / (A1 * r^2)
0.1975 = r^4
Take the fourth root of both sides:
∛(0.1975) = ∛(r^4)
0.6481 = r
Substitute the value of r back into equation (1) to find A1:
324 = A1 * (0.6481)^2
324 = A1 * 0.4200
A1 = 771.43
Therefore, the common ratio (r) is approximately 0.6481 and the value of A1 is approximately 771.43.
2. For the second problem:
Given:
A4 = 32/3
A8 = 54
Using the formula, we can set up two equations:
A4 = A1 * r^(4-1) ---> 32/3 = A1 * r^3 ----------(3)
A8 = A1 * r^(8-1) ---> 54 = A1 * r^7 ----------(4)
Divide equation (4) by equation (3):
(54) / (32/3) = (A1 * r^7) / (A1 * r^3)
5.0625 = r^4
Take the fourth root of both sides:
∛(5.0625) = ∛(r^4)
1.7188 = r
Substitute the value of r back into equation (3) to find A1:
32/3 = A1 * (1.7188)^3
32/3 = A1 * 6.5285
A1 = 1.032
Therefore, the common ratio (r) is approximately 1.7188 and the value of A1 is approximately 1.032.
To find the common ratio and the value of A1, we can use the formula for the nth term of a geometric sequence:
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 term number.
Let's solve each question step by step:
1. A3 = 324 and A7 = 64:
Using the formula, we can write the following equations:
A3 = A1 * r^(3-1) ...(1)
A7 = A1 * r^(7-1) ...(2)
Dividing equation (2) by equation (1), we get:
A7 / A3 = (A1 * r^(7-1)) / (A1 * r^(3-1))
64 / 324 = r^(6)
(2/9) = r^(6)
Taking the 6th root of both sides, we find:
r = (2/9)^(1/6) ≈ 0.783
Now, we can substitute the value of r back into equation (1) to find A1:
A3 = A1 * r^(3-1)
324 = A1 * (0.783)^(2)
324 = A1 * 0.613
A1 = 324 / 0.613
A1 ≈ 528.660
Therefore, the common ratio (r) is approximately 0.783, and the value of A1 is approximately 528.660.
2. A4 = 32/3 and A8 = 54:
Using the formula, we can write the following equations:
A4 = A1 * r^(4-1) ...(3)
A8 = A1 * r^(8-1) ...(4)
Dividing equation (4) by equation (3), we get:
A8 / A4 = (A1 * r^(8-1)) / (A1 * r^(4-1))
54 / (32/3) = r^(7)
54 * (3/32) = r^(7)
(81/32) = r^(7)
Taking the 7th root of both sides, we find:
r = (81/32)^(1/7) ≈ 1.191
Now, we can substitute the value of r back into equation (3) to find A1:
A4 = A1 * r^(4-1)
32/3 = A1 * (1.191)^(3)
32/3 = A1 * 1.617
A1 = (32/3) / 1.617
A1 ≈ 6.171
Therefore, the common ratio (r) is approximately 1.191, and the value of A1 is approximately 6.171.