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An arithmetic sequence is a list of numbers where each term is obtained by adding a constant value, called the common difference, to the preceding term. For example, consider the sequence 6, 15, 24, 33, 42, .... Here, the common difference is 9 because each term increases by 9. The formula for finding the n-th term of an arithmetic sequence is:

\[ a_n = a_1 + (n-1)d \]

where \(a_n\) is the n-th term, \(a_1\) is the first term, and \(d\) is the common difference. Let's break it down with an example: if we have the sequence 10, _, 40, and we need to find the missing term, we observe that the common difference is

\[ d = (40 - 10)/3 = 10. \]

Thus, the sequence becomes 10, 20, 30, 40.

A geometric sequence is a list of numbers where each term is obtained by multiplying the preceding term by a constant value, called the common ratio. For example, in the sequence 6, 12, 24, 48, 96, ..., the common ratio is 2 because each term is multiplied by 2. The formula for finding the n-th term of a geometric sequence is:

\[ a_n = a_1 \times r^{(n-1)} \]

where \(a_n\) is the n-th term, \(a_1\) is the first term, and \(r\) is the common ratio. For instance, if we have the sequence 10, _, 40 and need to find the missing term, we determine that the common ratio is

\[ r = \frac{40}{10} = 2. \]

Thus, the sequence is 10, 20, 40.

The common difference is a crucial element in arithmetic sequences. It is the number added to each term to get the next term. To find the common difference, we subtract the first term from the second term. The common difference (\(d\)) can be positive, negative, or zero, which influences the nature of the sequence:

• If \(d\) is positive, the sequence increases.
• If \(d\) is negative, the sequence decreases.
• If \(d\) is zero, all terms in the sequence are the same.

Let's revisit the sequence 2, _, 50. From the exercise solution, the common difference is \[ d = \frac{50 - 2}{3} = 16. \] Thus, our arithmetic sequence will be 2, 18, 34, 50.

The common ratio is fundamental in geometric sequences. It is the factor by which we multiply each term to obtain the following term. To find the common ratio (\(r\)), we divide the second term by the first term. Similar to the common difference, the sign and value of the common ratio determine the sequence's behavior:

• If \(r\) is greater than 1, the sequence grows.
• If \(r\) is between 0 and 1, the sequence shrinks.
• If \(r\) is negative, the sequence alternates sign.

For instance, in the sequence 2, _, 50, we find the common ratio is \[ r = \frac{50}{2} = 25. \] Therefore, the geometric sequence is 2, 8, 50.

Sequence analysis involves identifying whether a sequence is arithmetic or geometric and then determining the common difference or common ratio. This helps in predicting future terms of the sequence. Steps for sequence analysis include:

• Identify the type of sequence: Check the difference or ratio between terms.
• Calculate the common value: Subtract (for arithmetic) or divide (for geometric) terms.
• Predict future terms: Use the identified pattern to find missing terms.

Let’s analyze the sequence 4, _, 36. If it’s an arithmetic sequence, given the first and last terms, we find the common difference as: \[ d = \frac{36 - 4}{3} = 32. \] Therefore, the arithmetic sequence is 4, 20, 36. Similarly, for a geometric sequence, the common ratio is: \[ r = \frac{36}{4} = 9. \] Thus, the sequence is 4, 8, 16, 36.