Final answer:
The sum of the digits of the number created by adding 6, 66, 666, and so on until the last number with 100 digits is actually 6. Each '6' contributes to the sum, and after multiplying the series sum by 6, the final sum's digits are added to find that the sum of the digits is 6.
Step-by-step explanation:
The sum of the digits of the number 6+66+666+6666+.....+N where the last number contains 100 digits involves a pattern of repeated digits of 6. If we break this down, each term has one more '6' than the previous term, up until the last term with 100 digits. In each term, the digit '6' appears as many times as the number of digits in the term. So, the first term has 1 digit of 6, the second term has 2 digits of 6, and so on, until the last term with 100 digits of 6. To find the sum of all these '6's, we simply need to sum the number of times '6' appears across all terms.
We know that '6' will appear 1 + 2 + 3 + ... + 100 times. This is a sum of an arithmetic series. The sum of the first n natural numbers is given by the formula n(n+1)/2. Applying this formula, with n=100, we get 100(100+1)/2 which equals 5050. Since each '6' is worth 6 points, we multiply 5050 by 6 to get 30300. The sum of the digits is the sum of all these '6's, and since each one is a separate digit, we simply add them up to get the final sum.
However, we need to recognize that the question is asking us for the sum of the digits of this calculated sum, which is 3 + 0 + 3 + 0 + 0. Adding these values, we get 3+3=6. So the sum of the digits of the number 6+66+666+...+N where N has 100 digits of 6 is 6.