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The general term of an arithmetic sequence is a formula used to find any term in the sequence without listing all the previous ones. It is especially helpful when you need to find, say the 100th term, without manually calculating each of the preceding terms. In arithmetic sequences, the sequence is defined by a constant difference between consecutive terms.

The formula for the general term, denoted as \(a_n\), is expressed as:

• \(a_n = a_1 + (n - 1) \times d\)

Here's what the terms represent:

• \(a_n\): The term you wish to find.
• \(a_1\): The first term of the sequence.
• \(d\): The common difference between consecutive terms.
• \(n\): The position of the term in the sequence.

Understanding this formula helps you directly find any term in the sequence by substituting these values into the formula, making calculations more efficient and less prone to error compared to manual addition.

The common difference in an arithmetic sequence is the amount by which each term increases to get to the next term. This value stays constant throughout the sequence, which is why it's called 'common'. It is a significant component in forming the arithmetic sequence and is denoted by \(d\).

To find the common difference, you can subtract any term from the following term:

• \(d = a_{n+1} - a_n\)

In the example provided, the common difference \(d\) is given as 4. This implies that every term is 4 more than the previous term. Consistently applying this difference allows the sequence to extend infinitely, with each step predictable and systematic. Recognizing the common difference is essential for using the general formula of an arithmetic sequence.

The formula for an arithmetic sequence is essential in finding the general term, thereby allowing any specific term in the sequence to be calculated directly. The arithmetic sequence formula is given as:

• \(a_n = a_1 + (n - 1) \times d\)

Here is how you apply this formula by plugging in the numbers:- You start with the first term \(a_1\). In our context, this is 15.- The common difference \(d\) is the step between terms, and it's 4 in this example.
To find any term \(a_n\) of the sequence, like the formula for \(a_n\) already provided:

• \(a_n = 15 + (n - 1) \times 4\)
• Simplifying gives: \(a_n = 15 + 4n - 4\)
• And further simplification provides: \(a_n = 4n + 11\)

This specific formula can directly find the value of any term without needing preceding terms, marking the efficiency of this arithmetic sequence formula in problem-solving.