Recursive Formula For Geometric Sequence A_n = 6 * (-1/4)^(n-1)

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In the realm of mathematics, sequences play a crucial role, particularly geometric sequences. Understanding these sequences is essential for various applications, from calculating compound interest to modeling population growth. Today, we delve into the specifics of geometric sequences, focusing on how to derive a recursive formula from a given explicit formula. Our primary example is the explicit formula a_n = 6 * (-1/4)^(n-1). We will break down the process step-by-step, ensuring a comprehensive understanding of the concepts involved. This exploration will not only provide the answer to the question but also enhance your grasp of sequence formulation and manipulation.

Defining Geometric Sequences and Their Formulas

Before diving into the recursive formula, let's establish a solid foundation by defining geometric sequences and the formulas associated with them. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor, often referred to as the common ratio. This is in contrast to arithmetic sequences, where terms increase by a constant difference.

The explicit formula for a geometric sequence allows you to calculate any term directly by plugging in its position (n) in the sequence. The general form of an explicit formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

Where:

  • a_n is the nth term of the sequence.
  • a_1 is the first term of the sequence.
  • r is the common ratio.
  • n is the position of the term in the sequence.

The recursive formula, on the other hand, defines a term in the sequence based on the preceding term(s). It consists of two parts:

  1. The initial term(s), usually a_1.
  2. A rule for finding a_n using a_(n-1).

The general form of a recursive formula for a geometric sequence is:

a_1 = [initial term] a_n = a_(n-1) * r

Where:

  • a_1 is the first term.
  • a_n is the nth term.
  • a_(n-1) is the (n-1)th term (the term before the nth term).
  • r is the common ratio.

Understanding these formulas is crucial for converting between explicit and recursive forms and for solving problems related to geometric sequences.

Deconstructing the Given Explicit Formula: a_n = 6 * (-1/4)^(n-1)

Our task is to find the recursive formula corresponding to the given explicit formula: a_n = 6 * (-1/4)^(n-1). To accomplish this, we need to identify the first term (a_1) and the common ratio (r) from the explicit formula.

By comparing the given explicit formula with the general form a_n = a_1 * r^(n-1), we can directly extract these values.

The first term, a_1, is the coefficient multiplying the exponential term. In this case, a_1 = 6. This means the first number in our sequence is 6.

The common ratio, r, is the base of the exponential term. Here, r = -1/4. This indicates that each subsequent term in the sequence is obtained by multiplying the previous term by -1/4. The negative sign implies that the terms will alternate in sign, and the fraction suggests that the magnitude of the terms will decrease.

Having identified a_1 and r, we now have the key ingredients to construct the recursive formula.

Constructing the Recursive Formula

Now that we have identified the first term (a_1 = 6) and the common ratio (r = -1/4), we can construct the recursive formula. Recall that a recursive formula has two parts: the initial term and the recursive rule.

The first part is straightforward: we simply state the first term, which we found to be a_1 = 6. This serves as the starting point for the sequence.

The second part involves defining the recursive rule, which describes how to find any term a_n based on the previous term a_(n-1). In a geometric sequence, this is done by multiplying the previous term by the common ratio. Therefore, the recursive rule is:

a_n = a_(n-1) * r

Substituting the value of r we found earlier, we get:

a_n = a_(n-1) * (-1/4)

Combining the initial term and the recursive rule, the complete recursive formula for the given geometric sequence is:

{ a_1 = 6 a_n = a_(n-1) * (-1/4) }

This formula tells us that the sequence starts with 6, and each subsequent term is -1/4 times the previous term.

Verifying the Recursive Formula

To ensure the accuracy of our derived recursive formula, it is essential to verify it. We can do this by generating the first few terms of the sequence using both the explicit and recursive formulas and comparing the results. If the terms match, it confirms that our recursive formula is correct.

Let's generate the first four terms using the explicit formula a_n = 6 * (-1/4)^(n-1):

  • For n = 1: a_1 = 6 * (-1/4)^(1-1) = 6 * (-1/4)^0 = 6 * 1 = 6
  • For n = 2: a_2 = 6 * (-1/4)^(2-1) = 6 * (-1/4)^1 = 6 * (-1/4) = -3/2
  • For n = 3: a_3 = 6 * (-1/4)^(3-1) = 6 * (-1/4)^2 = 6 * (1/16) = 3/8
  • For n = 4: a_4 = 6 * (-1/4)^(4-1) = 6 * (-1/4)^3 = 6 * (-1/64) = -3/32

Now, let's generate the first four terms using the recursive formula:

{ a_1 = 6 a_n = a_(n-1) * (-1/4) }

  • a_1 = 6 (given)
  • a_2 = a_1 * (-1/4) = 6 * (-1/4) = -3/2
  • a_3 = a_2 * (-1/4) = (-3/2) * (-1/4) = 3/8
  • a_4 = a_3 * (-1/4) = (3/8) * (-1/4) = -3/32

Comparing the terms generated by both formulas, we see that they match perfectly: 6, -3/2, 3/8, -3/32. This confirms that our recursive formula is indeed correct.

Common Mistakes and How to Avoid Them

When working with recursive formulas, especially in geometric sequences, there are several common mistakes that students often make. Recognizing these pitfalls can help you avoid errors and strengthen your understanding.

  1. Forgetting the Initial Term: A recursive formula is incomplete without the initial term(s). The initial term serves as the starting point for the sequence. Failing to include it will result in an undefined sequence. Always remember to explicitly state a_1 (and potentially other initial terms if needed).

  2. Incorrectly Identifying the Common Ratio: The common ratio is the constant factor by which each term is multiplied to obtain the next term. Ensure you correctly identify this value from the explicit formula or by dividing any term by its preceding term. Pay close attention to the sign of the ratio, as a negative ratio will result in alternating signs in the sequence.

  3. Mixing Up Explicit and Recursive Formulas: Explicit formulas directly calculate a term based on its position, while recursive formulas define a term based on previous terms. Confusing these two types of formulas can lead to significant errors. Practice converting between the two forms to solidify your understanding.

  4. Arithmetic Errors: When calculating terms using the recursive formula, be careful with arithmetic operations, especially when dealing with fractions or negative numbers. Double-check your calculations to ensure accuracy.

  5. Not Verifying the Formula: It's always a good practice to verify your derived recursive formula by generating a few terms and comparing them with the terms obtained from the explicit formula (if available). This helps catch any errors in your formulation.

By being mindful of these common mistakes, you can improve your accuracy and confidence in working with recursive formulas.

Conclusion

In conclusion, we have successfully derived the recursive formula for the geometric sequence with the explicit formula a_n = 6 * (-1/4)^(n-1). We began by understanding the fundamental concepts of geometric sequences, explicit formulas, and recursive formulas. We then deconstructed the given explicit formula to identify the first term (a_1 = 6) and the common ratio (r = -1/4). Using these values, we constructed the recursive formula:

{ a_1 = 6 a_n = a_(n-1) * (-1/4) }

We verified the formula by generating the first four terms using both the explicit and recursive formulas, confirming their equivalence. Finally, we discussed common mistakes to avoid when working with recursive formulas.

Understanding how to convert between explicit and recursive formulas is a crucial skill in mathematics, particularly in the study of sequences and series. It allows for a deeper comprehension of the patterns and relationships within these mathematical structures. This knowledge is not only valuable for academic pursuits but also for real-world applications involving exponential growth and decay, financial modeling, and other areas where sequences play a significant role.

By mastering the techniques outlined in this discussion, you will be well-equipped to tackle similar problems and confidently explore the fascinating world of geometric sequences.