Problem 37 - Algebra 2

Question

Find the missing positive term in each geometric sequence. $$ 64, \quad, 4, \dots $$

Step-by-Step Solution

Answer

The missing term in the geometric sequence is 64.

Step 1: Determine the Common Ratio

Given the sequence \(64, \quad, 4\), we can determine the common ratio by dividing the known adjacent terms. That is, \(r = \frac{4}{64} = \frac{1}{16}\). So the common ratio \(r\) is \(\frac{1}{16}\).

Step 2: Use the Ratio to Find the Missing Term

Now that we know the ratio, we can find the missing term by multiplying the first term by the ratio. Given the first term \(a = 64\), and \(r = \frac{1}{16}\), the missing term \(a_2\) can be calculated as \(a * r = 64 * \frac{1}{16} = 4\). Thus, the second term \(a_2\) is 4.

Step 3: Validate the Sequence

To ensure we found the correct term, check if it correctly fits into the sequence making it a geometric sequence. Taking the sequence \(64, 4, 4\), if we divide sequential terms we should get the same ratio throughout. Thus, \(\frac{4}{64} = \frac{1}{16}\) and \(\frac{4}{4} = 1\). Indeed, this is not a geometric sequence. Our second term should have been calculated by using the ratio on the second term instead of the first.

Step 4: Correct Mistake and Find the True Missing Term

After recognizing our mistake, we realize we have to find the second term by multiplying \(4\) (the third term) by \(16\) (the inverse of our ratio), because in a geometric sequence, each term is the previous term multiplied by \(r\). That means, \(a_2 = 4 * 16 = 64\). Therefore, the missing term is 64, making the correct sequence: \(64, 64, 4\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a fundamental aspect that determines how the sequence progresses. It is the factor by which we multiply each term to obtain the next term in the sequence.

• To find this ratio, simply divide any term in the sequence by the preceding term.
• The result should be consistent across the sequence.

For our example sequence, which starts with the terms 64 and 4, the common ratio is calculated as the ratio of the second term to the first term. That means \[ r = \frac{4}{64} = \frac{1}{16}.\]This common ratio \( r = \frac{1}{16} \) is essential for understanding how each term decreases by a factor of 16. Recognizing and calculating this ratio allows us to determine any missing terms and confidently predict future terms in a sequence.

Missing Term

Finding a missing term in a geometric sequence involves understanding and utilizing the common ratio. Once you have this ratio, the process becomes straightforward.

• Start with the first known term and the common ratio.
• Multiply the known term by the ratio to locate the next term in the sequence.

In our example, we initially started with the sequence 64, 4, and needed to find a term between them. With a miscalculated approach, there was confusion. Correctly applying the ratio, \[ a_2 = \text{next known term} \times \frac{1}{\text{ratio}},\]where 4 is known as the last term, helps verify the missing term is actually 64. Thus, our sequence should correctly read 64, 64, 4. This illustrates the importance of precise calculations when determining the missing term.

Sequence Validation

Validating a sequence ensures that all terms adhere to the rules of being a consistent geometric progression. Here's how you can validate effectively:

• Check if dividing any pair of consecutive terms yields the consistent common ratio.
• The sequence should maintain this ratio across all parts.

In the original attempt for our sequence, the initial calculation of terms didn't satisfy a valid geometric sequence, as dividing 4 by 4 equaled 1, instead of the common ratio \( \frac{1}{16} \).
To resolve this, we must identify whether there has been a mistake and correct it accordingly. By validating a geometric sequence, we avoid errors and confirm its integrity, knowing each term logically and mathematically follows its predecessor, maintaining the common ratio.