Find the geometric series whose second term is 1.5 and the fifth term is 0.1875
a + a r + a r^2 + a r^3 + a r^4 ....+a r^(n-1)
so
a r = 1.5
a r^4 = .1875
a = 1.5/r
(1.5/r) r^4 = .1875
r^3 = .125
r = .5
a = 1.5/.5 = 3
so
3 + 1.5 + .75 ......
To find the geometric series, we need to determine the common ratio between the terms.
Let's denote the first term of the series as 'a' and the common ratio as 'r'.
Given that the second term is 1.5, we can express it as:
a * r = 1.5 ---(1)
Given that the fifth term is 0.1875, we can express it as:
a * r^4 = 0.1875 ---(2)
To find the values of 'a' and 'r', we will solve the system of equations formed by equations (1) and (2).
Dividing equation (2) by equation (1), we get:
(r^4) = (0.1875) / (1.5)
Simplifying the right side, we have:
r^4 = 0.125
Taking the fourth root of both sides, we get:
r = √(0.125)
Since r must be positive for a geometric series, we take the positive square root:
r = 0.5
Substituting this value of r into equation (1), we can solve for 'a':
a * (0.5) = 1.5
Simplifying the equation, we have:
a = 1.5 / 0.5
a = 3
Therefore, the geometric series is:
3, 1.5, 0.75, 0.375, 0.1875
The common ratio, r, is 0.5, and the first term, a, is 3.