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.