Derive the formula for the sum of a geometric series S_n = (a(rⁿ − 1))/(r-1) given the first term a = 30 and n=12 complete calculations.

Let $S_n$ be the sum of the first $n$ terms of a geometric series.
Let $a$ be the first term and $r$ be the common ratio.
The terms of the series are $a, ar, ar^2, \dots, ar^{n-1}$.
The sum $S_n$ can be written as:

$$S_n = a + ar + ar^2 + \dots + ar^{n-1} \quad (1)$$

Multiplying every term in Equation (1) by $r$:

$$rS_n = ar + ar^2 + ar^3 + \dots + ar^n \quad (2)$$

Subtracting the first equation from the second equation:

$$rS_n - S_n = (ar + ar^2 + ar^3 + \dots + ar^n) - (a + ar + ar^2 + \dots + ar^{n-1})$$

Notice that most terms on the right-hand side cancel out:

$$S_n(r-1) = ar^n - a$$

Factor $S_n$ from the left side and $a$ from the right side:

$$S_n(r-1) = a(r^n - 1)$$

Assuming $r \neq 1$, divide both sides by $(r-1)$:

$$S_n = \frac{a(r^n - 1)}{r-1}$$

This formula shows that the total value (sum) of a geometric series after $n$ terms is given by $\frac{a(r^n - 1)}{r-1}$.