Fill in the blanks so that
___, 8, ___, ___, 27, ___, ...
is
a. an arithmetic sequence.
b. a geometric sequence.
c. a sequence that is neither arithmetic nor geometric, for which you are able to write a general term.
A sequence is a list of numbers or terms that follow a specific pattern or rule. 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.
To find the nth term in a sequence, you must first identify the pattern or rule that the sequence follows. Once you have determined the pattern, you can use it to find the missing term by plugging in the value of n. For example, if the sequence is 2, 4, 6, 8, 10, the pattern is adding 2 to the previous term. So, the nth term would be 2n.
In an arithmetic sequence, each term is found by adding a constant number to the previous term. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3. In a geometric sequence, each term is found by multiplying the previous term by a constant number. For example, 2, 6, 18, 54, 162 is a geometric sequence with a common ratio of 3.
A series is convergent if the sum of its terms approaches a finite value as the number of terms increases. It is divergent if the sum of its terms approaches infinity as the number of terms increases. To determine if a series is convergent or divergent, you can use various tests such as the nth term test, integral test, or comparison test.
Sequences and series are used in many fields, including finance, computer science, and physics. In finance, compound interest and annuities can be modeled using geometric sequences and series. In computer science, sequences are used in algorithms and data structures. In physics, sequences and series are used to model physical phenomena and calculate probabilities in quantum mechanics.