In a geometric sequence ,the 3rd term is 24 and 5th term is 3 .write down first for sequences
ar^2 = 24
ar^4 = 3
r^2 = 1/8
not sure what "the first for sequences" is supposed to mean.
To find the first four terms of the geometric sequence, we need to determine the common ratio and then use this to calculate the terms.
The formula for the nth term of a geometric sequence is given by:
Tn = a * r^(n-1),
where Tn represents the nth term, a is the first term, r is the common ratio, and n is the term number.
We are given the 3rd term as 24 and the 5th term as 3:
T3 = 24
T5 = 3
Using the formula, we can write the following equations:
a * r^2 = 24 (equation 1)
a * r^4 = 3 (equation 2)
Dividing equation 2 by equation 1:
(a * r^4) / (a * r^2) = 3 / 24
r^2 = 1/8
Taking the square root of both sides:
r = ± sqrt(1/8)
Since a geometric sequence cannot have a negative common ratio, we take the positive square root:
r = sqrt(1/8)
r = 1/2
Now that we know the common ratio, we can substitute it into one of the given equations to find the value of the first term, a.
Using equation 1 with r = 1/2:
a * (1/2)^2 = 24
a * (1/4) = 24
a = 24 * 4
a = 96
So, the first term of the geometric sequence is 96.
To find the first four terms, we substitute the values of a and r into the formula for the nth term:
T1 = 96 * 1^(1-1) = 96 * 1^0 = 96 * 1 = 96
T2 = 96 * 1^(2-1) = 96 * 1^1 = 96 * 1 = 96
T3 = 96 * 1^(3-1) = 96 * 1^2 = 96 * 1 = 96
T4 = 96 * 1^(4-1) = 96 * 1^3 = 96 * 1 = 96
Therefore, the first four terms of the geometric sequence are:
96, 96, 96, 96
To write down the first four terms of the geometric sequence, let's first find the common ratio (r) of the sequence.
In a geometric sequence, the ratio between any two consecutive terms is constant. We can use this property to find the common ratio.
Given that the 3rd term is 24 and the 5th term is 3, we can set up two equations:
24 = a * r^2 (equation 1)
3 = a * r^4 (equation 2)
Here, a represents the first term of the geometric sequence.
Dividing equation 1 by equation 2, we get:
24/3 = (a * r^2) / (a * r^4)
8 = r^2 / r^4
8 = 1 / r^2
Taking the reciprocal of both sides, we have:
1/8 = r^2
Taking the square root of both sides, we find:
r = ±√(1/8) = ±1/√8 = ±1/2√2 = ±1/(2√2)
Since the common ratio cannot be negative (as it would result in negative terms), we can discard the negative sign. Therefore, the common ratio is:
r = 1/(2√2) = (1/2)√2
Now that we have the common ratio, we can find the first term (a) by substituting the known values into equation 1:
24 = a * ((1/2)√2)^2
24 = a * (1/4) * 2
24 = a/2
a = 48
Thus, the first term (a) of the geometric sequence is 48, and the common ratio (r) is (1/2)√2.
Now, we can write down the first four terms of the sequence:
1st term: a = 48
2nd term: a * r = 48 * (1/2)√2 = 24√2
3rd term: a * r^2 = 48 * ((1/2)√2)^2 = 48 * (1/4) * 2 = 24
4th term: a * r^3 = 48 * ((1/2)√2)^3 = 48 * (1/8) * √2 = 6√2
Therefore, the first four terms of the geometric sequence are:
48, 24√2, 24, 6√2.
5 = 8
4 = 16
3 = 24
2 = 32
1 = 40