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In an arithmetic sequence, each term comes after the previous one by adding a fixed number called the common difference. This feature is what makes arithmetic sequences predictable and simple to analyze. To find the common difference, you subtract the first term from the second term. In our given sequence, the first term is 4 and the second term is 7. Calculating this difference:

• Subtract the first term from the second term: 7 - 4 = 3.

Thus, the common difference, denoted as \(d\), is 3. This means every term in the sequence increases by 3 from the previous term. Keeping the common difference in mind is key for understanding how arithmetic sequences are structured and helps us predict future terms. Besides adding to our computational skills, this concept is a fundamental building block for higher-level mathematics.

The magic of arithmetic sequences lies in their ability to be described by a simple formula. This formula gives us any term in the sequence without needing to list all the terms leading up to it. The sequence formula is:

• \( a_n = a_1 + (n-1) \cdot d \)

This formula tells us:- \(a_n\) stands for any term in the sequence.- \(a_1\) is the first term.- \(n\) represents the position of the term you want to find.- \(d\) is the common difference.By inputting the values of \(a_1\), \(n\), and \(d\), you can calculate any specific term swiftly. This formula is incredibly useful because it saves you from computing each term consecutively, especially in sequences with a vast number of terms. It's like having a shortcut for arithmetic sequences, ensuring you reach the desired result quickly and accurately.

Term calculation involves using the sequence formula to pinpoint a particular term in the arithmetic sequence. Let's see how you can calculate the fifth term in our sequence.We start with:

• First term \(a_1 = 4\)
• Common difference \(d = 3\)
• Position \(n = 5\)

Using the sequence formula \(a_n = a_1 + (n-1) \cdot d\), we plug in the values:

• \(a_5 = 4 + (5-1) \cdot 3\)

Simplifying this gives:

• \(a_5 = 4 + 4 \cdot 3 = 4 + 12 = 16\)

Hence, the fifth term is 16. Term calculation not only provides practice in applying the sequence formula, but also enriches our comprehension of arithmetic sequences overall. By mastering this skill, any student can navigate through the seemingly complex world of mathematical sequences with confidence and clarity.