42. Sequences and geometric series

Página 42

The sequence and geometric series are fundamental concepts in mathematics and are often required in the ENEM test. Understanding these concepts can help students solve a variety of problems, from simple math questions to complex physics and engineering problems.

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed constant, called the ratio. The ratio can be any real number. The sequence 2, 4, 8, 16, 32 is an example of a geometric sequence where the ratio is 2.

To find the nth term of a geometric sequence, we use the formula a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the ratio, and n is the position of the term in the sequence.

For example, to find the 5th term of the geometric sequence 2, 4, 8, 16, 32, we substitute a_1 for 2, r for 2, and n for 5 in the formula to get a_5 = 2 * 2^(5-1) = 2 * 16 = 32, which is the 5th term of the sequence.

A geometric series is the sum of the terms of a geometric sequence. For example, the series 2 + 4 + 8 + 16 + 32 is a geometric series where the ratio is 2.

To find the sum of the first n terms of a geometric series, we use the formula S_n = a_1 * (1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a_1 is the first term, r is the ratio and n is the number of terms.

For example, to find the sum of the first 5 terms of the geometric series 2 + 4 + 8 + 16 + 32, we substitute a_1 for 2, r for 2, and n for 5 in the formula to get S_5 = 2 * (1 - 2 ^5) / (1 - 2) = 2 * (-31) / (-1) = 62, which is the sum of the first 5 terms of the series.

It is important to note that the formula for the sum of the first n terms of a geometric series is only valid if the ratio r is different from 1. If the ratio is 1, the series is a constant series and the sum of the first n terms is simply n times the first term.

Also, if the ratio is greater than 1 or less than -1, the geometric series is a divergent series, which means that the sum of the terms tends to infinity as n tends to infinity. If the ratio is between -1 and 1, the geometric series is a convergent series, which means that the sum of the terms tends to a fixed number as n tends to infinity.

In summary, geometric sequences and series are important mathematical concepts that are often required in the ENEM test. Understanding these concepts can help students solve a variety of problems and do well on the test.

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