find the general term (un), of the geometric sequence which has :
u7=24 and u15= 384
thank u
The ratio of u15 and u7 is 16, which is 2^4
The term number (n) change from u7 to u15 is 8. A factor of 16 change would result from each term being higher by 2^(1/2) or sqrt2.
So try un = C*2^(n/2)
Evaluate C as follows:
When n = 7
24 = C*2^(7/2)
24 = C*8 sqrt2. Therefore
C = 3/sqrt2
un = (3/sqrt2)*2^(n/2)
Check: u15 = [3/2^(1/2)]*2^15/2
= 3* 2^7 = 384
It works
Ops
To find the general term (un) of a geometric sequence, we need to determine the common ratio (r) first.
We are given two terms of the sequence: u7 = 24 and u15 = 384.
We can use these values to find the common ratio:
u7 = u1 * r^(7-1) (where u1 is the first term of the sequence)
24 = u1 * r^6
u15 = u1 * r^(15-1)
384 = u1 * r^14
Divide the second equation by the first equation to eliminate u1:
384 / 24 = (u1 * r^14) / (u1 * r^6)
16 = r^8
Taking the square root of both sides:
√16 = √(r^8)
4 = r^4
Now we have the common ratio (r = 4).
To find the general term (un), we can use the formula:
un = u1 * r^(n-1)
Substituting the known values:
u7 = u1 * r^6
24 = u1 * 4^6
24 = u1 * 4096
Divide both sides by 4096:
24 / 4096 = u1
Simplify:
3 / 512 = u1
Therefore, the general term (un) of the geometric sequence is:
un = (3 / 512) * r^(n-1)
un = (3 / 512) * 4^(n-1)
To find the general term (un) of a geometric sequence, you need the value of any two terms in the sequence. In this case, we have u7 = 24 and u15 = 384.
The general term of a geometric sequence can be found using the formula:
un = u1 * r^(n-1)
where un is the nth term, u1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
To find the common ratio (r), we can use the fact that the ratio of any two consecutive terms in a geometric sequence is constant.
First, we can find the value of u1. To do this, we need to find the common ratio (r) by using the given terms:
r = u15 / u7 = 384 / 24 = 16
Now, we can use the value of r to find u1:
u1 = u7 / r^(7-1) = 24 / 16^6
Simplifying the equation:
u1 = 24 / (2^4)^6 = 24 / 2^24
Thus, the general term of the sequence is:
un = (24 / 2^24) * 16^(n-1)
Note: The value of u1 can also be simplified further, but it depends on whether the question requires a simplified form or not.