17. Numerical sequences and series

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Numeric sequences and series are important concepts in the study of mathematics and are often addressed in the Enem test. A numeric sequence is an ordered list of numbers where each number has a specific position. A numerical series, on the other hand, is the sum of the terms of a numerical sequence.

Numeric Sequences

A numerical sequence can be defined in several ways. A common way to define a sequence is through an explicit formula that gives the nth term of the sequence. For example, the sequence of natural numbers can be defined by the formula n, where n is the position of the term in the sequence.

Sequences can also be defined by a recurrence rule, which provides a way to calculate each term based on previous terms. For example, the sequence of Fibonacci numbers is defined by the recurrence rule F(n) = F(n-1) + F(n-2), with initial values ​​F(1) = 1 and F(2) = 1 .

There are two main types of numerical sequences: arithmetic and geometric. An arithmetic sequence is one in which the difference between consecutive terms is constant. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic sequence with a common difference of 2. A geometric sequence is one in which the quotient of consecutive terms is constant. For example, the sequence 3, 6, 12, 24, 48 is a geometric sequence with a common ratio of 2.

Numeric Series

A numerical series is the sum of the terms of a numerical sequence. For example, the series 1 + 2 + 3 + 4 + 5 is the sum of the first five terms of the sequence of natural numbers.

There are several ways to sum a series. One way is to simply add the terms one at a time. However, for series with many terms or with terms that are not whole numbers, this may be impractical. Another way is to use a formula for the sum of a series, if one exists.

For example, the sum of the first n terms of an arithmetic sequence can be found by the formula n/2 * (a + l), where a is the first term, l is the last term, and n is the number of terms. The sum of the first n terms of a geometric sequence can be found by the formula a * (r^n - 1) / (r - 1), where a is the first term, r is the common ratio and n is the number of terms.

Importance for the Enem

Understanding numerical sequences and series is essential for success in the Enem test. These concepts are often used in problems involving number patterns, exponential growth and decay, and mathematical modeling. Furthermore, the ability to manipulate sequences and series can be useful in other areas of mathematical study, such as calculus and linear algebra.

In summary, numerical sequences and series are important concepts that are frequently tested in the Enem exam. Understanding these concepts and knowing how to work with them can help students solve a variety of math problems and succeed on the test.

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1818. Arithmetic and geometric progressions