Geometric Sequences, Series & Financial Applications

A geometric sequence multiplies rather than adds: each term is obtained from the previous one by multiplying by a fixed number called the common ratio, denoted $r$. Starting from a first term $u_1$, the sequence runs $u_1,\; u_1 r,\; u_1 r^2,\; u_1 r^3,\dots$, and the nth term is $$u_n = u_1 r^{\,n-1}.$$ The exponent is $n-1$ for the same reason as before: it takes $n-1$ multiplications to reach the $n$th term. A sequence is geometric when consecutive ratios are constant, so $r = \dfrac{u_{n+1}}{u_n}$ is the same for every $n$. If $|r| > 1$ the terms grow in magnitude; if $0 < |r| < 1$ they shrink toward zero; and if $r$ is negative the terms alternate in sign.

The geometric series sums the terms. For $r \neq 1$, $$S_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}.$$ The two ve