Sum of numbers and/or digits 1-100

What is the sum of the DIGITS of 1 + 2 + 3 +...+ 100?

A good way to approach this problems is to consider smaller problems 
and look for a patterm.

Let's first ask, what is the sum of the digits 0-9? Clearly, this is 
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9, which equals 45.  

Next question: What is the sum of the digits of the numbers 10-19? 
We know that these numbers all have a "1" in the tens place, and a 
number 0-9 in the units place. There are ten of these numbers, so 
there are ten 1's and the digits 0, 1, 2, 3, 4...9. So, the sum of 
these numbers must be 10x1 + (0 + 1 +...+ 9).

What about the sum of the digits of the numbers 20-29? Again, we need 
to add ten 2's and the digits 0-9, which is 10x2 + (0 + 1 +...+ 9).

Do you see the pattern? So, when we add up the digits of all numbers 
0-99, the expression is:

       (0 + 1 +...+ 9) +              The numbers 0-9
10x1 + (0 + 1 +...+ 9) +              The numbers 10-19
10x2 + (0 + 1 +...+ 9) +              The numbers 20-29
10x3 + (0 + 1 +...+ 9) +              The numbers 30-39
10x4 + (0 + 1 +...+ 9) +              The numbers 40-49
10x5 + (0 + 1 +...+ 9) +              The numbers 50-59
10x6 + (0 + 1 +...+ 9) +              The numbers 60-69
10x7 + (0 + 1 +...+ 9) +              The numbers 70-79
10x8 + (0 + 1 +...+ 9) +              The numbers 80-89
10x9 + (0 + 1 +...+ 9)                And, finally, the numbers 90-99.

Thank goodness we can simplify this! From the distributive law of 
multiplication, we know that 

(10x1 + 10x2 +...+ 10x9) = 10(1 + 2 ...9). 

We also know that there are ten additions of (0 + 1 +...+ 9), 
which is another way of saying that there are ten times 
(0 + 1 +...+ 9),  10(0 + 1 +...+ 9). 

So, the sum of digits of the numbers 1-99 is:

10(0 + 1 +...+ 9) + 10(0 + 1 +...+ 9) = 20(0 + 1 + 2...9)
               
Since (0 + 1 +...+ 9) = 45, this equals: 
                              20 x 45 = 900

We add the digit 1 from "100" because we want the sum of digits 1-100, 
and the final answer is 901.