Summing All Digits from 1 to 1 Billion

[Qwark]

Homework Statement

Find the sum of all the digits in the integers from 1 to 1 000 000 000.
Ex: 2378:2+3+7+8=20

Homework Equations

We're not supposed to use equations for the sum of a series.

The Attempt at a Solution

I really don't have much of a clue as to how to start, so any help pointing me in the right direction would be great.

 

[3trQN]

Start by trying it for a smaller set, try 1 to 10.

 

[Qwark]

I tried it out by adding the digits of the integers on opposing sides of 1 and a billion and got a final answer of 412 999 999 921. I'm not sure if this is anywhere near the right answer though.

 

[Dr Transport]

There is a trick you can use...Think about it some more...

Hint: it involves the first digit and the last digit.

 

[Edgardo]

Hello Qwark,

could you elaborate how you got to the answer of 412 999 999 921?


My calculations yielded:

4,05*10^{10}+1= 40 500 000 001

I first calculated the sum of the digits of the numbers from:
1 to 9 then
10 to 99 then
100 to 999 then
1.000 to 9.999 then
10.000 to 99.999 then
100.000 to 999.999 then
1.000.000 to 9.999.999 then
10.000.000 to 99.999.999 then
100.000.000 to 999.999.999 then
1.000.000.000

1 to 9 yields 45
10 to 99 yields 855 and so on.

Then I added the sums and got
40 500 000 001

Please check if you get the same result.

I don't know how Dr. Transport's trick looks like but my solution
contains an iteration.

 

[Qwark]

I can tell that I made a false asumption when I got the answer 412 999 999 921, I was trying to find a pattern by adding the digits of opposing numbers, and made a false assumption based on the first few answers and jumped to conclusions. I'm curious as to how you got the 4.05 in your calculation.

 

[Edgardo]

Try calculating the sum of the digits of the numbers from 1 to 9. What do you get?

After that, calculate the same for the numbers from 10 to 99. What do you get?

Do it also for the numbers from 100 to 999. What do you get?