Geometrical Progression.

Geometrical Progression. A series of quantities is said to be in geometrical progression when the ratio of each term to the preceding is the same for all the terms—i.e. when any term is equal to the product of the preceding term and a factor which is the same throughout the series. This constant ratio or factor is termed the common ratio. For example, the numbers 2, 4, 8, 16, &c., and also the terms , are both examples of geometrical progression or series. The sum of such a series is obtained as follows: Let $a$ be the first term, $n$ the number of the terms whose sum, $s$, is required, and let $r$ be the common ratio. Then $s\; =\; a\; +\; ar\; +\; ar^2\; +\; \backslash dots\; +\; ar^\{n-1\}$; also from multiplication of both sides of this equation by $r$, $sr\; =\; ar\; +\; ar^2\; +\; ar^3\; +\; \backslash dots\; +\; ar^n$. Subtraction of the former from the latter expression gives $sr\; -\; s\; =\; ar^n\; -\; a$; or $s(r\; -\; 1)\; =\; a(r^n\; -\; 1)$, and hence $s\; =\; \backslash frac\{a(r^n\; -\; 1)\}\{r\; -\; 1\}$.