The 3rd Term Is 24 ---> ar^2 = 24
The 5th Term Is 3 ---> ar^4 = 3
divide them
r^2 = 3/24 = 1/8
r = 1/2โ2
a(1/64) = 24
a = 24*64 = ...
now form your terms
The 5th Term Is 3 ---> ar^4 = 3
divide them
r^2 = 3/24 = 1/8
r = 1/2โ2
a(1/64) = 24
a = 24*64 = ...
now form your terms
r = 1/โ8 = โ2/4
ar^2 = 24
a = 24*8 = 192
So the sequence is
192, 48โ2, 24, 6โ2, 3, ...
Let's call the first term in the sequence a.
Using the formula for the nth term in a geometric sequence:
5th term = a * r^4 = 3 [since 5-1=4]
3rd term = a * r^2 = 24 [since 3-1=2]
Now, let's solve these two equations for a and r.
(a * r^4) / (a * r^2) = 3 / 24
r^2 = 24/3 = 8
Taking the square root of both sides, we find:
r = โ8 = 2โ2
Now that we have the value of r, we can find the first term (a) by substituting it back into the 3rd term equation:
a * (2โ2)^2 = 24
4a = 24
a = 6
So, the first term (a) is 6, and the common ratio (r) is 2โ2.
The first four terms of the geometric sequence are:
a = 6
ar = 6 * 2โ2 = 12โ2
ar^2 = 12โ2 * 2โ2 = 48
ar^3 = 48 * 2โ2 = 96โ2
Therefore, the first four terms in the sequence are 6, 12โ2, 48, and 96โ2.
Given that the third term is 24 and the fifth term is 3, we can set up the following equations:
24 = a * r^2 -- (equation 1)
3 = a * r^4 -- (equation 2)
Here, "a" represents the first term in the sequence.
To solve for "a" and "r," we divide equation 2 by equation 1:
(3 / 24) = (a * r^4) / (a * r^2)
Simplifying, we get:
(1/8) = r^2
Taking the square root of both sides:
โ(1/8) = r
Simplifying further:
r = 1/2
Now, we can substitute this value of "r" back into equation 1 to solve for "a":
24 = a * (1/2)^2
24 = a * 1/4
Multiplying both sides by 4:
96 = a
Therefore, the first term in the sequence (a) is 96, and the common ratio (r) is 1/2.
To write down the first four terms, we can use the formula for the nth term of a geometric sequence:
a_n = a * r^(n-1)
Substituting the values we found:
a_1 = 96 * (1/2)^(1-1) = 96 * 1 = 96
a_2 = 96 * (1/2)^(2-1) = 96 * 1/2 = 48
a_3 = 96 * (1/2)^(3-1) = 96 * 1/4 = 24
a_4 = 96 * (1/2)^(4-1) = 96 * 1/8 = 12
Therefore, the first four terms in the sequence are: 96, 48, 24, 12.