Insert five geometric means between 0.125 and 8.
I know one set is 0.25, 0.5, 1, 2, 4
What would be the other set?
.125 , .125r, .125r^2, .125r^3, .125r^4, .125r^5, 8
.125r/.125 = 8/.125r^5
r = 64/r^5
r^6 = 64
r = ± 2
set 1:
.25, .5, 1, 2, 4
set 2: if r = -2
-.25, .5, -1, 2, -4
I think you have the answer. The question said 5 means and didn't say 5 different sets of means. I can't think of another way that you can get from .125 to 8.
Write the geometric sequence that has five geometric means between 1and 15,625.
1, 5, 20, 125, 615, 3,125, 15,625, …
1, 5, 25, 125, 625, 3,125, 15,625, …
1, 5, 25, 125, 620, 3,250, 15,625, …
1, 5, 25, 125, 652, 3,350, 15,625, …
To find the other set of geometric means between 0.125 and 8, we need to understand the concept of geometric progression.
Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
In this case, the first term is 0.125, and the last term is 8. To find the common ratio, we divide the last term by the first term: 8/0.125 = 64.
Now, to find the five geometric means, we need to repeatedly multiply the first term (0.125) by the common ratio (64). The formula to find the nth term of a geometric sequence is:
an = a1 * r^(n-1)
where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Let's calculate the other set of geometric means:
2nd term: a2 = a1 * r^1 = 0.125 * 64 = 8
3rd term: a3 = a1 * r^2 = 0.125 * (64^2) = 512
4th term: a4 = a1 * r^3 = 0.125 * (64^3) = 32768
5th term: a5 = a1 * r^4 = 0.125 * (64^4) = 2097152
6th term: a6 = a1 * r^5 = 0.125 * (64^5) = 134217728
Therefore, the other set of geometric means between 0.125 and 8 is: 0.25, 0.5, 1, 2, 4, 8, 512, 32768, 2097152, 134217728.