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Chapter 7: Problem 14

Find the sum of all the four-digit positive integers that are evenly divisible by 5 .

Short Answer

Expert verified
The sum of all four-digit positive integers that are evenly divisible by 5 is determined by first finding the lowest and highest four-digit multiples of 5 as 1000 and 9995. Next, calculate the number of terms in the sequence, which is \(1 + \frac{(9995 - 1000)}{5} = 1901\). Finally, use the arithmetic series formula to find the sum: \(\frac{1901}{2} \times (1000 + 9995) = 950.5 \times 10995 = 10443999.5\). Therefore, the sum of all the four-digit positive integers that are evenly divisible by 5 is 10,443,999.

Step by step solution

01

Identify the lowest and highest four-digit multiples of 5

In this step, we need to find the lowest and highest four-digit integers that are multiples of 5. The first four-digit integer is 1000, and we can check that 1000 is not a multiple of 5. The lowest possible four-digit multiple of 5 will be the next multiple of 5 after that. The last four-digit integer is 9999, and we can find the highest four-digit multiple of 5 by rounding it down to the nearest multiple of 5.
02

Determine the number of terms in the sequence

Now that we have identified the lowest and highest four-digit multiples of 5, we can find the number of terms in this arithmetic sequence. The number of terms in an arithmetic sequence can be calculated using the formula: Number of terms = \(1 + \frac{(Last\,term - First\,term)}{Common\,Difference}\)
03

Find the sum of the arithmetic series

Next, we will find the sum of the arithmetic series, which can be calculated using the arithmetic series formula: Sum = \(\frac{Number\,of\,terms}{2}\) 脳 \((First\,term + Last\,term)\)

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