Explicit Formula For The Sequence 2, 10, 50, 250, 1250

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Finding the explicit formula for a sequence is a fundamental problem in mathematics, particularly in the study of sequences and series. An explicit formula allows us to directly calculate any term in the sequence without needing to know the preceding terms. This is incredibly useful for analyzing the behavior of sequences and making predictions about future terms. In this article, we will explore how to determine the explicit formula for the sequence 2, 10, 50, 250, 1250, ... and discuss the underlying concepts involved.

Understanding Sequences and Explicit Formulas

Before diving into the specifics of the given sequence, it’s essential to understand what a sequence is and what an explicit formula represents. A sequence is an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is called a term. For example, in the sequence 2, 10, 50, 250, 1250, ..., each number is a term, and the position of the term in the sequence is typically denoted by n, where n is a positive integer (1, 2, 3, ...).

An explicit formula, also known as a closed-form expression, provides a direct way to calculate the n-th term of a sequence, denoted as a_n, using a formula that depends only on n. This contrasts with a recursive formula, which defines a term based on the preceding terms. Explicit formulas are highly desirable because they offer a straightforward method for finding any term without having to compute all the preceding terms. For instance, if we have an explicit formula a_n = f(n), we can find the 100th term by simply plugging n = 100 into the formula.

Explicit formulas are crucial for several reasons. Firstly, they provide a compact and efficient representation of a sequence. Instead of listing out terms, we can define the entire sequence with a single formula. Secondly, explicit formulas allow us to analyze the long-term behavior of a sequence. We can use them to determine whether a sequence converges, diverges, or oscillates. Thirdly, they are essential in various applications, such as in calculus, where we study the limits of sequences and series, and in computer science, where sequences are used in algorithms and data structures. Understanding and deriving explicit formulas is, therefore, a cornerstone of mathematical analysis.

Analyzing the Given Sequence: 2, 10, 50, 250, 1250, ...

To find the explicit formula for the sequence 2, 10, 50, 250, 1250, ..., the first step is to identify the pattern or rule that governs the sequence. By observing the sequence, we can see that each term is obtained by multiplying the previous term by 5. This indicates that the sequence is a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant value, known as the common ratio.

To confirm this, let’s calculate the ratio between consecutive terms:

  • 10 / 2 = 5
  • 50 / 10 = 5
  • 250 / 50 = 5
  • 1250 / 250 = 5

The ratio between consecutive terms is consistently 5, confirming that the sequence is indeed geometric with a common ratio of 5. The first term of the sequence, denoted as a_1, is 2. Now that we have identified the common ratio and the first term, we can proceed to derive the explicit formula for this geometric sequence.

Geometric sequences are characterized by a constant ratio between successive terms. This constant ratio, often denoted by r, is a key factor in determining the behavior of the sequence. In our case, the common ratio r = 5 indicates that the terms will grow rapidly as n increases. The first term a_1 serves as the starting point for the sequence, and each subsequent term is obtained by multiplying the previous term by the common ratio. Recognizing these properties is crucial for constructing the explicit formula, which will encapsulate the relationship between the term number n and the value of the term a_n. This understanding not only helps in solving this specific problem but also provides a foundation for analyzing other geometric sequences and series.

Deriving the Explicit Formula for a Geometric Sequence

The general form of an explicit formula for a geometric sequence is given by:

a_n = a_1 r^(n-1)

where:

  • a_n is the n-th term of the sequence,
  • a_1 is the first term of the sequence,
  • r is the common ratio, and
  • n is the term number.

In our case, we have identified that a_1 = 2 and r = 5. Substituting these values into the general formula, we get:

a_n = 2 * 5^(n-1)

This is the explicit formula for the sequence 2, 10, 50, 250, 1250, .... This formula allows us to calculate any term in the sequence directly. For example, to find the 6th term, we would substitute n = 6:

a_6 = 2 * 5^(6-1) = 2 * 5^5 = 2 * 3125 = 6250

The explicit formula we derived is a powerful tool for understanding and working with geometric sequences. It provides a concise way to represent the entire sequence and allows us to perform calculations efficiently. The process of deriving the formula involves identifying the key characteristics of the sequence, such as the first term and the common ratio, and then applying the general form of the explicit formula. This method is applicable to any geometric sequence, making it a valuable technique in mathematical problem-solving. By understanding the underlying principles and practicing with different examples, one can become proficient in finding explicit formulas for geometric sequences.

Evaluating the Given Options

Now that we have derived the explicit formula for the sequence, we can compare it with the given options to identify the correct one. The options provided are:

A. a_n = 2 + (5)^n B. a_n = 5(2)^(n-1) C. a_n = 2(5)^n D. a_n = 2(5)^(n-1)

Our derived formula is a_n = 2 * 5^(n-1). Comparing this with the options, we can see that option D matches our derived formula exactly.

Let's evaluate each option to further illustrate why option D is the correct one:

  • Option A: a_n = 2 + (5)^n. This formula is incorrect because it involves addition rather than multiplication of the common ratio. If we plug in n = 1, we get a_1 = 2 + 5 = 7, which does not match the first term of our sequence (2).
  • Option B: a_n = 5(2)^(n-1). This formula does not represent the given sequence because it uses a base of 2 instead of 5 for the exponential term. Plugging in n = 1, we get a_1 = 5(2)^0 = 5, which is also incorrect.
  • Option C: a_n = 2(5)^n. This formula is close but not quite right. It has the correct base of 5 and the correct leading coefficient of 2, but the exponent should be (n-1) rather than n. If we plug in n = 1, we get a_1 = 2(5)^1 = 10, which is not the first term of our sequence.
  • Option D: a_n = 2(5)^(n-1). This formula matches our derived formula. Plugging in n = 1, we get a_1 = 2(5)^(1-1) = 2(5)^0 = 2, which is the first term. Plugging in n = 2, we get a_2 = 2(5)^(2-1) = 2(5)^1 = 10, which is the second term. This formula correctly generates the sequence.

Therefore, the correct explicit formula for the sequence 2, 10, 50, 250, 1250, ... is option D: a_n = 2(5)^(n-1).

Conclusion

In conclusion, finding the explicit formula for a sequence involves identifying the pattern and expressing it in a mathematical form that allows direct calculation of any term. For the sequence 2, 10, 50, 250, 1250, ..., we identified it as a geometric sequence with a first term of 2 and a common ratio of 5. Using the general formula for the explicit form of a geometric sequence, we derived the formula a_n = 2(5)^(n-1). By comparing this formula with the given options, we confirmed that option D is the correct explicit formula.

Understanding explicit formulas is crucial for analyzing sequences and making predictions about their behavior. The ability to derive and apply these formulas is a valuable skill in mathematics and has applications in various fields, including computer science, engineering, and finance. By mastering the techniques discussed in this article, you can confidently tackle problems involving sequences and series and deepen your understanding of mathematical patterns and relationships. The process of identifying patterns, formulating explicit formulas, and verifying their correctness is a fundamental aspect of mathematical thinking and problem-solving.