HELP NEEDED . HARD MATH PROBLEM!!!!?

Question

Observe the expression 1 + 11 + 101 + 1001 + 10001 + … + 1000…0001 where the last term has one hundred zeroes. When the expression is calculated and written as a single positive integer, what will be the sum of the digits?

Answer 1

Best Answer - Chosen by Voters

Think of it this way
Suppose instead of that sum, you had

1 + 10 + 100 + 1000 + 10000 + 100000 + 1000000 + 10000000 + etc...
then your final number would be

11111111111....11111111111111111111111… (102 digit number)

What else are you adding to that number?

11 = 10 + 1
101 = 100 + 1
1001 = 1000 + 1
1000 (with 100 digits of zeros) 01 = 10000 (with 100 digits of zeros) + 1

So your final number is

11111111111111111111111111111111111111… (lots of 1s) 11111 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... (101 times)


or

111111111111111111111111111111111111 (lots of 1s) + 101

so your final number is

11111111111111111111111....11111111111…

This number has 102 digits, so 100 1's and 2 2's, so the sum of the digits is 104.

Answer 2

98 1's followed by 2 zeros

Answer 3

you just add the number that is 102 consecutive 1's to the number 102
and you get 99 1's followed by a 2 another 1 and a three...
summing those digits up gets you 105