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The sum of an arithmetic series is a central concept in working with arithmetic sequences. An arithmetic series is simply the sum of the terms in an arithmetic sequence. To find this sum, we use a specific formula that requires knowledge of both the first and last terms, as well as the number of terms in the sequence. The formula is: \[ S_n = \frac{n}{2} \cdot (a + a_n) \] Where:

• \(S_n\) is the sum of the series
• \(n\) is the number of terms
• \(a\) is the first term
• \(a_n\) is the last term

To use this formula, calculate \(\frac{n}{2}\) first, which effectively finds the average of the first and last terms when multiplied by \(n\). Then, add the first term \(a\) to the last term \(a_n\), and multiply this sum by \(\frac{n}{2}\). In our exercise, by plugging in the values \(n = 16\), \(a = 4\), and \(a_n = 64\), the sum was calculated to be 544.

The common difference is what makes an arithmetic sequence unique. It is the fixed amount that each term increases by to get to the next term in the sequence. You can find the common difference, \(d\), by subtracting the first term from the second term. For example, in our sequence 4, 8, 12, ... , 64, we can find \(d\) as follows:

• Subtract the first term from the second term: \(8 - 4 = 4\).

This number \(d = 4\) is what separates each number from the next in our sequence. Understanding the common difference helps determine other elements of the sequence, such as the total number of terms and assists in calculating the series sum. If the common difference is consistent, the sequence is indeed arithmetic.

Finding the number of terms in an arithmetic sequence is a crucial step in solving problems related to sequences and series. The formula to determine the number of terms \(n\) when given the last term \(a_n\), first term \(a\), and common difference \(d\) is based on the n-th term formula for an arithmetic sequence: \[ a_n = a + (n-1) \cdot d \] By rearranging this formula, you solve for \(n\):

• Replace \(a_n\), \(a\), and \(d\) with their respective values.
• Simplify the equation to find \(n\).
• For example, use \(64 = 4 + (n-1) \cdot 4\) and solve: \(64 = 4n\), then \(n = 16\).

This shows that there are 16 terms from the first term 4 to the last term 64, considering our common difference of 4. Knowing \(n\) is essential for computing the total sum of the sequence.