The third term of a geometric series is 24 and the fourth term is 36. Determine the first term and common ratio.
T4/T3 = r = 36/24 = 3/2
ar^2 = 24, so
a = 24(4/9)
a = 32/3
To find the first term and common ratio of a geometric series, we can use the formulas:
nth term = a * r^(n-1)
Given that the third term of the series is 24 and the fourth term is 36, we can set up the following equations:
24 = a * r^(3-1)
36 = a * r^(4-1)
Let's solve these equations simultaneously to determine the values of a (the first term) and r (the common ratio).
First, divide the second equation by the first equation to eliminate 'a':
36/24 = (a * r^(4-1))/(a * r^(3-1))
1.5 = r^(4-1)/r^(3-1)
1.5 = r^3/r^2
Next, simplify the equation by multiplying both sides by r^2:
1.5 * r^2 = r^3
Now, rearrange the equation:
r^3 - 1.5 * r^2 = 0
Factor out 'r^2':
r^2 * (r - 1.5) = 0
This equation has two possible solutions:
1) r^2 = 0 -> r = 0 (not a valid solution for a geometric series)
2) r - 1.5 = 0 -> r = 1.5
Now, substitute the value of r = 1.5 into one of the original equations:
24 = a * (1.5)^(3-1)
Simplify the equation:
24 = a * (1.5)^2
Multiply 1.5^2:
24 = a * 2.25
Now, divide both sides by 2.25 to solve for 'a':
a = 24 / 2.25
a = 10.67
Therefore, the first term (a) is approximately 10.67 and the common ratio (r) is 1.5.