Finding The 6th Term In The Geometric Sequence 1, 0.5, 0.25

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In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding geometric sequences is a fundamental concept in algebra and calculus, with applications ranging from financial calculations (like compound interest) to physics (like radioactive decay). In this article, we will delve into how to find a specific term in a geometric sequence, focusing on the given example: 1, 0.5, 0.25, ... Specifically, we aim to identify and calculate the 6th term of this sequence.

Understanding Geometric Sequences

Before we dive into solving the problem, it's crucial to understand the fundamental concepts of geometric sequences. A geometric sequence is defined by its first term (a) and its common ratio (r). The common ratio is the factor by which each term is multiplied to obtain the next term. Mathematically, the nth term (an) of a geometric sequence can be expressed using the formula:

an = a * r(n-1)

Where:

  • an is the nth term of the sequence.
  • a is the first term of the sequence.
  • r is the common ratio.
  • n is the term number (the position of the term in the sequence).

To effectively use this formula, we first need to identify the first term (a) and the common ratio (r) from the given sequence. In the sequence 1, 0.5, 0.25, ..., the first term (a) is clearly 1. To find the common ratio (r), we can divide any term by its preceding term. For instance, dividing the second term (0.5) by the first term (1) gives us 0.5. Similarly, dividing the third term (0.25) by the second term (0.5) also yields 0.5. This confirms that the common ratio (r) for this sequence is 0.5. Once we have identified the first term and the common ratio, we can proceed to calculate any term in the sequence using the formula mentioned above. In our case, we want to find the 6th term, so we will substitute the values of a, r, and n (which is 6) into the formula. This systematic approach ensures that we can accurately determine any term in a geometric sequence, making it a powerful tool in various mathematical and real-world applications.

Identifying the First Term and Common Ratio

In the geometric sequence provided, 1, 0.5, 0.25, ..., the first step in finding the 6th term is to accurately identify the first term and the common ratio. These two values are the foundational elements that define the entire sequence and are crucial for calculating any specific term. The first term, denoted as a, is simply the initial value in the sequence. In this case, the first term is clearly 1. This value serves as the starting point from which all subsequent terms are generated. The common ratio, denoted as r, is the constant factor by which each term is multiplied to obtain the next term. To find the common ratio, we can divide any term in the sequence by its preceding term. This process ensures that we are capturing the multiplicative relationship that governs the sequence. For instance, we can divide the second term (0.5) by the first term (1), which gives us 0.5. Alternatively, we can divide the third term (0.25) by the second term (0.5), which also results in 0.5. The consistency of this result confirms that the common ratio for this sequence is indeed 0.5. This means that each term in the sequence is half of the previous term. Accurately determining the first term and the common ratio is essential because these values are directly used in the formula for finding the nth term of a geometric sequence. Without these values, it would be impossible to calculate any specific term, including the 6th term that we are aiming to find. By carefully identifying these values, we set the stage for the next step in the process, which involves applying the geometric sequence formula.

Applying the Formula for the nth Term

Once we have identified the first term (a) as 1 and the common ratio (r) as 0.5, the next step is to apply the formula for the nth term of a geometric sequence. This formula, an = a * r(n-1), is the key to finding any specific term in the sequence. In our case, we want to find the 6th term, which means we need to calculate a6. To do this, we will substitute the values we have identified into the formula. Specifically, a will be replaced with 1, r will be replaced with 0.5, and n will be replaced with 6. This substitution transforms the general formula into a specific equation that we can solve for the 6th term. The equation becomes:

a6 = 1 * (0.5)(6-1)

This equation represents the 6th term as the product of the first term (1) and the common ratio (0.5) raised to the power of (6-1), which is 5. The exponent (n-1) in the formula reflects the fact that the first term is not multiplied by the common ratio, but each subsequent term is. By following the order of operations, we first calculate the exponent and then perform the multiplication. This ensures that we arrive at the correct value for the 6th term. The application of this formula is not only crucial for solving this specific problem but also provides a general method for finding any term in any geometric sequence. By understanding and applying this formula, we can efficiently determine the value of any term, making it a powerful tool in various mathematical and practical contexts.

Calculating the 6th Term

Having set up the equation a6 = 1 * (0.5)(6-1), we now proceed with calculating the 6th term. This involves evaluating the expression on the right-hand side of the equation, following the order of operations. First, we simplify the exponent: (6 - 1) equals 5. So, the equation becomes:

a6 = 1 * (0.5)5

Next, we need to calculate (0.5)5, which means 0.5 raised to the power of 5. This can be calculated as 0.5 * 0.5 * 0.5 * 0.5 * 0.5. Multiplying these values together, we get:

(0.5)5 = 0.03125

Now, we substitute this value back into the equation:

a6 = 1 * 0.03125

Finally, we perform the multiplication: 1 multiplied by 0.03125 equals 0.03125. Therefore, the 6th term of the geometric sequence is 0.03125. This result indicates that the terms in the sequence are decreasing, as the common ratio is less than 1. The 6th term is significantly smaller than the first term, reflecting the effect of the common ratio on the sequence. The process of calculating the 6th term demonstrates the practical application of the geometric sequence formula. By following the steps carefully and performing the calculations accurately, we can determine any term in the sequence. This skill is valuable in various mathematical contexts and real-world applications where geometric sequences are used to model phenomena such as exponential decay or compound interest. Understanding how to calculate specific terms in a geometric sequence allows us to make predictions and analyze patterns, enhancing our problem-solving abilities.

The 6th term of the geometric sequence 1, 0.5, 0.25, ... is 0.03125.

In conclusion, by carefully identifying the first term and the common ratio, applying the formula for the nth term of a geometric sequence, and performing the necessary calculations, we have successfully determined the 6th term of the geometric sequence 1, 0.5, 0.25, ... to be 0.03125. This process underscores the importance of understanding the fundamental concepts of geometric sequences and the power of the formula an = a * r(n-1). This formula allows us to find any term in a geometric sequence, given the first term, the common ratio, and the term number. The steps involved in solving this problem provide a clear and systematic approach that can be applied to other geometric sequence problems. First, we identify the first term (a) and the common ratio (r). Then, we substitute these values, along with the term number (n), into the formula. Finally, we perform the calculations, following the order of operations, to find the value of the desired term. The ability to find specific terms in a geometric sequence has numerous applications in various fields, including finance, physics, and computer science. For example, geometric sequences can be used to model compound interest, radioactive decay, and the growth of populations. Therefore, mastering this concept is essential for anyone studying mathematics or related fields. The solution to this problem not only provides a specific answer but also reinforces a general method for solving geometric sequence problems, enhancing our understanding and problem-solving skills in mathematics. By following this approach, we can confidently tackle a wide range of problems involving geometric sequences and their applications.