The 6th and the 13th term of a GP are 24 and 3 by 16 respectively find the sequence
T13/T6 = ar^12/ar^5 = r^7
(3/16)/24 = 1/128 = (1/2)^7
so, ...
To find the sequence of a geometric progression (GP), we need to determine the common ratio (r). Once we have the common ratio, we can use it to find the first term (a).
Step 1: Find the common ratio (r)
We are given that the 6th term of the GP is 24 and the 13th term is 3/16. The formula to find the nth term of a GP is given by:
an = a * r^(n-1)
Using this formula, we can set up two equations using the given information:
24 = a * r^5 -------------- Equation 1
3/16 = a * r^12 -------------- Equation 2
Step 2: Solve the equations to find the values of a and r
To solve these two equations simultaneously, we can divide Equation 2 by Equation 1:
(3/16) / 24 = (a * r^12) / (a * r^5)
3/16 / 24 = r^(12-5)
1/128 = r^7
Taking the 7th root of both sides:
r = (1/128)^(1/7)
r = 1/2
Step 3: Find the first term (a)
Substituting the value of r into Equation 1:
24 = a * (1/2)^5
24 = a * 1/32
24 * 32 = a
a = 768
So, the first term (a) of the geometric progression is 768, and the common ratio (r) is 1/2.
Step 4: Find the sequence
Now that we have the first term (a) and the common ratio (r), we can find the sequence by listing the terms:
a1 = 768
a2 = 768 * 1/2
a3 = (768 * 1/2) * 1/2
a4 = ((768 * 1/2) * 1/2) * 1/2
...
Continuing this pattern, we can find the values of the sequence.
To find the sequence of a geometric progression (GP), we need to know the first term (a) and the common ratio (r).
Given that the 6th term is 24 and the 13th term is 3/16, we can set up two equations based on the positions and values of these terms:
a * r^5 = 24 ① (since the 6th term is a * r^5)
a * r^12 = 3/16 ② (since the 13th term is a * r^12)
Now, we can solve these two equations simultaneously to find the values of a and r.
Divide equation ② by equation ① to eliminate "a" and solve for "r":
(a * r^12) / (a * r^5) = (3/16) / 24
r^7 = 3/16 * 1/24
r^7 = 1/128
Take the 7th root of both sides to solve for "r":
r = (1/128)^(1/7)
r ≈ 0.5
Substitute the value of "r" back into equation ① to solve for "a":
a * (0.5)^5 = 24
a * 0.03125 = 24
a ≈ 768
So, the first term (a) is approximately 768 and the common ratio (r) is approximately 0.5.
Now, we can generate the sequence by multiplying the first term by the common ratio repeatedly:
Sequence: 768, 384, 192, 96, 48, 24, 12, ...
Note: The sequence continues indefinitely with each term being half of the preceding term.