A sequence is a list of numbers or terms that follow a specific pattern or rule, whereas a series is the sum of the terms in a sequence. In other words, a series is the result of adding all the terms in a sequence together.
In order to determine the next term in a sequence, you need to identify the pattern or rule that the sequence follows. This can be done by looking at the differences between consecutive terms, the ratio between consecutive terms, or any other pattern that may be present. Once the pattern is identified, it can be used to predict the next term in the sequence.
The formula for finding the sum of a finite arithmetic series is (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term. Alternatively, the formula can also be written as (n/2)(2a + (n-1)d), where d is the common difference between consecutive terms.
A series is said to be convergent if the sum of its terms approaches a finite value as the number of terms in the series increases. This can be determined by applying various convergence tests, such as the ratio test, the integral test, or the comparison test. If the sum of the terms does not approach a finite value, the series is said to be divergent.
Sequences and series are used in various real-life applications, such as finance, physics, and computer science. In finance, they are used to calculate compound interest and to model stock market trends. In physics, they are used to model motion and radioactive decay. In computer science, they are used in algorithms and data structures.