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Find the sum of all positive integers less than $1000$ ending in $1$ or $2$ or $3$ or $4.$

 Mar 12, 2024
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We can find the sum of all positive integers less than 1000 by itself, and then subtract the sum of the numbers ending in 0, 5, 6, 7, 8, and 9.

 

The sum of all positive integers less than 1000 is $ \frac{1000 \cdot 999}{2} = 499500$.

 

The number of integers less than 1000 ending in 0 is 9 (from 10 to 900). The sum of an arithmetic series is the number of terms in the series times the average of the first and last term.

 

In this case, the sum of the integers ending in 0 is 9⋅(210+900​)=4050. We can do the same for the other digits 5 through 9.

 

Therefore, the sum of all integers less than 1000 ending in 1, 2, 3, or 4 is 499500−6⋅4050=499500−24300=475200​.

 Mar 12, 2024

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