Sum to n terms of GP is (r ≠ 1):

A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). Let the sum of the first $n$ terms be $S_n$: $$S_n = a + ar + ar^2 + \dots + ar^{n-1} \quad \text{--- (Equation 1)}$$ To derive the formula, multiply both sides of Equation 1 by the common ratio $r$: $$rS_n = ar + ar^2 + ar^3 + \dots + ar^n \quad \text{--- (Equation 2)}$$ Now, subtract Equation 1 from Equation 2: $$rS_n - S_n = (ar + ar^2 + \dots + ar^n) - (a + ar + \dots + ar^{n-1})$$ Notice that all terms cancel out except for the last term of Equation 2 and the first term of Equation 1: $$S_n(r - 1) = ar^n - a$$ Factor out $a$ on the right side: $$S_n(r - 1) = a(r^n - 1)$$ Divide both sides by $(r - 1)$, assuming $r \neq 1$: $$S_n = \frac{a(r^n - 1)}{r - 1}$$ Thus, the sum to $n$ terms of a GP is $\frac{a(r^n - 1)}{r - 1}$.