When you have a sequence of consecutive integers, the sum of the sequence can be found by multiplying the average of the first and last term by the number of terms. In this case, you have a sequence of five consecutive positive multiples of 7. The average of the first and last term would be (7 + 7*4) / 2 = 21.
So, the sum of the five consecutive positive multiples of 7 is:
Sum = Average * Number of Terms = 21 * 5 = 105.
Now, let's consider the possible values of the sum that can be obtained using this sequence. The minimum sum you can get is 105, which is when you use the sequence {7, 14, 21, 28, 35}. The maximum sum you can get is 105 + 4 * 7 = 133, which is when you use the sequence {28, 35, 42, 49, 56}.
Any integer between 105 and 133 (inclusive) can be expressed as the sum of five consecutive positive multiples of 7 using different sequences. Since both 105 and 133 are inclusive, there are a total of 133 - 105 + 1 = 29 integers between 1 and 2009 (inclusive) that can be expressed as the sum of five consecutive positive multiples of 7.
