1970 AHSME Problems/Problem 33
Problem
Find the sum of digits of all the numbers in the sequence 
.
Solution
[b]Solution 1[/b]
We can find the sum using the following method. We break it down into cases. The first case is the numbers 
to 
. The second case is the numbers 
to 
. The third case is the numbers 
to 
. The fourth case is the numbers 
to 
. And lastly, the sum of the digits in 
. The first case is just the sum of the numbers to which is, using 
, 
. In the second case, every number to is used 
times. times in the tens place, and times in the ones place. So the sum is just 
. Similarly, in the third case, every number to is used times in the hundreds place, 
times in the tens place, and times in the ones place, for a total sum of 
. By the same method, every number to is used times in the thousands place, 
times in the hundreds place, times in the tens place, and times in the ones place, for a total of 
. Thus, our final sum is 
[b]Solution 2[/b]
Consider the numbers from 
. We have 
digits and each has equal probability of being 
Our requested sum then is 
[i]Credit: Math1331Math [/i]
See also
| 1970 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 32 |
Followed by Problem 34 | |
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