To determine the first four terms of the geometric series, we need to find the common ratio (r) and the first term (a).
Let's use the given information to set up equations:
1. The sum of the first two terms of the series is 12:
a + ar = 12
2. The sum of the first three terms of the series is 62:
a + ar + ar^2 = 62
Now, we have a system of two equations with two variables. We can solve it to find the values of a and r.
First, let's solve the first equation for a:
a + ar = 12
a(1 + r) = 12
a = 12 / (1 + r)
Substitute this expression for a in the second equation:
(12 / (1 + r)) + (12 / (1 + r))r + (12 / (1 + r))r^2 = 62
To simplify this equation, multiply through by (1 + r):
12 + 12r + 12r^2 = 62(1 + r)
Rearrange the equation:
12r^2 + 12r - 62(1 + r) + 12 = 0
12r^2 + 12r - 62 - 62r + 12 = 0
12r^2 - 50r - 38 = 0
Now, we can solve this quadratic equation for r. Once we find the values of r, we can substitute them back into the first equation to find the corresponding values of a.
Using the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 12, b = -50, and c = -38:
r = (-(-50) ± √((-50)^2 - 4(12)(-38))) / 2(12)
r = (50 ± √(2500 + 1824)) / 24
r = (50 ± √(4324)) / 24
r ≈ (50 ± 65.71) / 24
Simplifying the two possible values of r:
r₁ ≈ (50 + 65.71) / 24 ≈ 115.71 / 24 ≈ 4.82
r₂ ≈ (50 - 65.71) / 24 ≈ -15.71 / 24 ≈ -0.65
Since a geometric series cannot have a negative common ratio, we discard r₂ and consider only r₁.
Now, substitute r = 4.82 back into the first equation:
a + ar = 12
a + 4.82a = 12
5.82a = 12
a ≈ 12 / 5.82 ≈ 2.06
So, the first term (a) is approximately 2.06, and the common ratio (r) is approximately 4.82.
To find the first four terms of the series, use the formula for the nth term of a geometric series:
an = a * r^(n-1), where n represents the term number.
Substituting the known values:
a₁ ≈ 2.06
a₂ ≈ 2.06 * 4.82^1 ≈ 2.06 * 4.82 ≈ 9.93
a₃ ≈ 2.06 * 4.82^2 ≈ 2.06 * 4.82 * 4.82 ≈ 46.91
a₄ ≈ 2.06 * 4.82^3 ≈ 2.06 * 4.82 * 4.82 * 4.82 ≈ 226.43
Therefore, the first four terms of the given geometric series are approximately:
2.06, 9.93, 46.91, 226.43.