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An arithmetic sequence, sometimes called an arithmetic progression, is a series of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3. When dealing with four-digit multiples of 5, we're looking at a specific type of arithmetic sequence where each number ends in a 0 or a 5 and is within the range 1000 to 9999.

To establish the arithmetic sequence for our problem, we first identify the smallest and largest four-digit numbers that are multiples of 5. We find that the series begins with 1000 (which is incorrect as 1000 is evenly divisible by 5) and ends with 9995. All the terms in this particular sequence will then be separated by a common difference of 5, since each consecutive multiple of 5 increases by this amount.

Four-digit multiples of 5 are numbers ranging from 1000 to 9999 that are divisible by 5. This means that their last digit must be either 0 or 5. To determine the correct lowest four-digit multiple of 5, we start at 1000, which is actually a multiple of 5; therefore, 1000 is the first term of our sequence. The sequence continues with 1005, 1010, 1015, and so on.

As we progress to higher numbers, we eventually come to the largest four-digit number divisible by 5, which is 9995. Understanding the range and characteristics of these numbers is crucial because it sets the foundation for determining the number of terms in the sequence and calculating the sum of all these numbers, a task typically encountered in mathematical exercises and real-world problems.

The sum of an arithmetic series is the total value obtained when all terms in the sequence are added together. To find this sum efficiently, especially when the series contains a large number of terms, we use a specific formula:Sum = \(\frac{Number\,of\,terms}{2}\) \times \((First\,term + Last\,term)\).In our scenario, after identifying the first term (1000) and the last term (9995), we apply the formula for the number of terms in the sequence:Number of terms = \(1 + \frac{(Last\,term - First\,term)}{Common\,Difference}\).Once we calculate the number of terms, we apply it to the sum formula to obtain the total sum of the arithmetic series of four-digit multiples of 5. This formula works because it essentially multiplies the average of the first and last term by the number of terms, giving us the overall sum without having to add each term individually.