1970 AHSME Problems/Problem 33
Problem
Find the sum of digits of all the numbers in the sequence .
Solution
Solution 1
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
Solution 2
Consider the numbers from . We have digits and each has equal probability of being Our requested sum then is // Credit: Math1331Math
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 32 |
Followed by Problem 34 | |
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