The first two terms in a geometric series is 12. The sum of the first three terms of the same series is 62. Determine the first four terms of the series.
To solve this problem, we need to use the formula for the sum of a geometric series and some algebraic manipulation.
Let's denote the first term of the series as 'a' and the common ratio as 'r'.
According to the problem, the first two terms are 12. This means that the first term is 12 and the second term is 12 * r.
So, we have the equations:
a = 12 (equation 1)
ar = 12r (equation 2)
Now, the formula for the sum of a geometric series is:
Sum = a * (1 - r^n) / (1 - r)
where 'Sum' is the sum of the first 'n' terms.
According to the problem, the sum of the first three terms is 62. So, we have:
a * (1 - r^3) / (1 - r) = 62
Now, we can substitute the value of 'a' from equation 1 into this equation:
12 * (1 - r^3) / (1 - r) = 62
To simplify this equation, let's multiply both sides by (1 - r):
12 * (1 - r^3) = 62 * (1 - r)
Simplifying further:
12 - 12r^3 = 62 - 62r
Rearrange this equation to have all terms on one side:
12r^3 - 62r + 50 = 0
Now we can solve this cubic equation to find the value(s) of 'r'. Once we find 'r', we can substitute it back into equations 1 and 2 to find the first four terms of the series.