 Geometric progression

A geometric progression is a sequence obtained by repeatedly multiplying by one fixed common ratio $r$:Here $a$ and $r$ may be real numbers or complex numbers. The recurrence is meaningful even when $a=0$ or $r=0$; expressing $r$ as a ratio of successive terms requires a nonzero denominator. As a polynomial convention, $r^0=1$, including at $r=0$.

For example, quantities $100,80,64,51.2,\dots$ retain four fifths of their preceding value at every step. The decrement changes, but the retained fraction does not. With the same initial value and common ratio $2$, the terms are $100,200,400,800,\dots$. This multiplicative structure is useful whenever the same proportional rule repeats: it turns many updates into the single exponentiation $r^n$. An arithmetic progression instead repeats an additive increment.

For nonzero $a$, the modulus of $a_n$ tends to zero when $|r|<1$, stays fixed when $|r|=1$, and grows without bound when $|r|>1$. For a negative real common ratio, successive nonzero terms also alternate in sign. Thus the terms $1,-1/2,1/4,-1/8,\dots$ decay in size while crossing zero at every step. Adding the terms gives a geometric series; the individual terms and their accumulated partial sums are different sequences.

A geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.