42. Sequences and geometric series

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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