## Mathematical Analysis## Sequences and Series |

## What is a geometric series? |

If we add the numbers in a geometric sequence, we end up with a geometric series. A geometric series is obtained when each term is determined from the preceding one by multiplying by a common ratio; there is a constant ratio between terms. For example, 1 + ½ + 1/4 + 1/8 + and so on, is a geometric series because each term is determined by multiplying the preceding term by ½. To find the sum of a geometric series, the formula is: Sum = *a*(*r*^{n} - 1) / (r - 1) or a(1 - *r*^{n}) / (1 - *r*), in which *a* is the first term, *r* is the common ratio, and *n* is the number of terms.

For example, to find the sum of the first six terms of a series represented by 2 + 6 + 18 + 54 + 162 + 486, define *a =* 2; *r =* 3; and *n =* 6. Substitute the numbers: Sum = 2(3^{6} - 1) / 3 - 1 = 729 - 1 = 728. We could also have determined this number based on the first few numbers, such as 2 + 6 + 18 + 54, as long as we knew the common ratio, the first number, and how many numbers in the series we wanted to add. This is something that can easily be determined based on just these four numbers.