Find the common ratio, r, for the geometric sequence that has a1= 100 and a8= 25/32
a8 = a1*r^7, so
r^7 = 100/(25/32) = 3200/25
an amusing little sequence, no?
r = 0.5
its
r^7= (25/32)/100
r = 7root [(25/32)/100}
r = 1/2 or 0.5
To find the common ratio, r, for a geometric sequence, we can use the formula:
an = a1 * r^(n-1)
Given that a1 = 100 and a8 = 25/32, we can write the equations:
a1 = 100 * r^(1-1) = 100 * r^0 = 100
a8 = 100 * r^(8-1) = 100 * r^7 = 25/32
Now, we can solve for r by dividing the equation for a8 by the equation for a1:
(100 * r^7) / 100 = (25/32)
Simplifying the equation:
r^7 = (25/32)
Taking the seventh root of both sides to isolate r:
r = (25/32)^(1/7)
So, the common ratio, r, for the given geometric sequence is (25/32)^(1/7).
To find the common ratio, r, of a geometric sequence, you can use the formula an = a1 * r^(n-1), where a1 is the first term, an is the nth term, and r is the common ratio.
We are given that a1 = 100 and a8 = 25/32. Let's use these values to find the common ratio.
For the first term (a1 = 100), we have n = 1. Plugging these values into the formula, we get:
100 = 100 * r^(1-1)
100 = 100 * r^0
100 = 100 * 1
100 = 100
Now, let's determine the value of the eighth term (a8 = 25/32). We have n = 8. Plugging these values into the formula, we get:
25/32 = 100 * r^(8-1)
25/32 = 100 * r^7
To find r, we need to isolate it on one side of the equation. Let's divide both sides of the equation by 100 first:
(25/32) / 100 = r^7
25/3200 = r^7
To simplify further, we can take the seventh root of both sides:
(r^7)^(1/7) = (25/3200)^(1/7)
r = (25/3200)^(1/7)
So, the common ratio (r) for the given geometric sequence is:
r = (25/3200)^(1/7)