A geometric series is a sequence of numbers where each consecutive number is multiplied by a constant value, known as the common ratio. The series follows the formula a + ar + ar^2 + ar^3 + ... + ar^n-1, where 'a' is the first term and 'r' is the common ratio.
The sum of a geometric series is the total value of all the terms in the series. It can be calculated using the formula S = a(1-r^n)/1-r, where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the series.
To find the sum of a finite geometric series, use the formula S = a(1-r^n)/1-r, where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the series. Simply plug in the values and solve for S.
The value of an infinite geometric series can be found using the formula S = a/(1-r), as long as the absolute value of 'r' is less than 1. If the absolute value of 'r' is greater than or equal to 1, the series will diverge and have no finite value.
Geometric series have various real-life applications, including compound interest, population growth, and radioactive decay. They can also be used in financial and investment planning, as well as in physics and engineering calculations.