18. Arithmetic and geometric progressions

Arithmetic and geometric progressions are fundamental topics in the study of mathematics and are often required in the Enem test. They are sequences of numbers that follow specific, predictable rules, making them a useful tool for solving a variety of mathematical problems.

In an Arithmetic Progression (AP), each term (except the first) is the sum of the previous term with a constant, called the ratio. For example, in a sequence 2, 4, 6, 8, each term is 2 more than the previous term, so the ratio is 2. The general formula for the nth term of an AP is a1 + (n-1)*r , where a1 is the first term, r is the ratio, and n is the number of the term we want to find.

To understand better, let's consider an AP where the first term is 3 and the ratio is 5. If we want to find the 4th term, we substitute in the formula: 3 + (4-1)*5 = 3 + 15 = 18. So the 4th term is 18.

The sum of the first n terms of an AP can also be calculated by the formula S = n/2 * (a1 + an), where S is the sum, n is the number of terms, a1 is the first term and an is the nth term. If, in the previous example, we wanted to add the first 4 terms, we would have S = 4/2 * (3 + 18) = 2 * 21 = 42.

In a Geometric Progression (GP), each term (except the first) is the product of the previous term by a constant, called the ratio. For example, in a sequence 2, 4, 8, 16, each term is 2 times the previous term, so the ratio is 2. The general formula for the nth term of a GP is a1 * r^(n-1), where a1 is the first term, r is the ratio, and n is the number of the term we want to find.

Considering a GP where the first term is 2 and the ratio is 3. If we want to find the 4th term, we substitute in the formula: 2 * 3^(4-1) = 2 * 27 = 54. Therefore, the 4th term is 54.

The sum of the first n terms of a GP can be calculated by the formula S = a1 * (1 - r^n) / (1 - r) if the ratio is different from 1. In the previous example, if we want to add the first 4 terms, we would have S = 2 * (1 - 3^4) / (1 - 3) = 2 * (-80) / -2 = 80.

It is important to note that arithmetic and geometric progressions are just two types of numerical sequences. There are many other types, but these are the most common and are often encountered in a variety of contexts, from pure mathematical problems to practical applications in physical sciences and economics.

Finally, it is essential to practice solving problems involving AP and PG to become familiar with their properties and be able to apply them effectively in the Enem test. Remember, math is a subject that requires continual practice to hone your skills and develop your intuition.

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