Multiples of 4 can be written as 4k, for some integer k.
The first three digits multiple of 4 is 4*25=100, and the last is 4*249=996.
So, the sum of all the three digits multiples of 4 is
![\displaystyle \sum_{k=25}^{249}4k = 4\sum_{k=25}^{249}k](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bk%3D25%7D%5E%7B249%7D4k%20%3D%204%5Csum_%7Bk%3D25%7D%5E%7B249%7Dk)
We know how to sum the first
integers, but we need to sum the first 249 integers starting from 25, so we can rewrite our sum using this little trick:
![\displaystyle \sum_{k=25}^{249}k = \sum_{k=1}^{249}k - \sum_{k=1}^{24}k](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bk%3D25%7D%5E%7B249%7Dk%20%3D%20%5Csum_%7Bk%3D1%7D%5E%7B249%7Dk%20-%20%5Csum_%7Bk%3D1%7D%5E%7B24%7Dk)
In other words, we're summing all the first 249 integers, but then we remove the first 24. As a result, we have the sum of all integers from 25 to 249:
![1+2+3+\ldots+248+249-(1+2+3+\ldots+23+24) = 25+26+27+\ldots+248+249](https://tex.z-dn.net/?f=1%2B2%2B3%2B%5Cldots%2B248%2B249-%281%2B2%2B3%2B%5Cldots%2B23%2B24%29%20%3D%2025%2B26%2B27%2B%5Cldots%2B248%2B249)
So, we have
![\displaystyle 4\sum_{k=25}^{249}k = 4\left(\sum_{k=1}^{249}k-\sum_{k=1}^{24}k\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%204%5Csum_%7Bk%3D25%7D%5E%7B249%7Dk%20%3D%204%5Cleft%28%5Csum_%7Bk%3D1%7D%5E%7B249%7Dk-%5Csum_%7Bk%3D1%7D%5E%7B24%7Dk%5Cright%29)
And in both cases we can use the formula
![\displaystyle \sum_{k=1}^{N}k=\dfrac{N(N+1)}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bk%3D1%7D%5E%7BN%7Dk%3D%5Cdfrac%7BN%28N%2B1%29%7D%7B2%7D)
And we have
![\displaystyle 4\left(\sum_{k=1}^{249}k-\sum_{k=1}^{24}k\right)=4\left(\dfrac{249\cdot 250}{2}-\dfrac{24\cdot 25}{2}\right)=2\cdot 249\cdot 250 - 2\cdot 24\cdot 25](https://tex.z-dn.net/?f=%5Cdisplaystyle%204%5Cleft%28%5Csum_%7Bk%3D1%7D%5E%7B249%7Dk-%5Csum_%7Bk%3D1%7D%5E%7B24%7Dk%5Cright%29%3D4%5Cleft%28%5Cdfrac%7B249%5Ccdot%20250%7D%7B2%7D-%5Cdfrac%7B24%5Ccdot%2025%7D%7B2%7D%5Cright%29%3D2%5Ccdot%20249%5Ccdot%20250%20-%202%5Ccdot%2024%5Ccdot%2025)
Which evaluates to
![2\cdot 249\cdot 250 - 2\cdot 24\cdot 25 = 124500-1200 = 123.300](https://tex.z-dn.net/?f=2%5Ccdot%20249%5Ccdot%20250%20-%202%5Ccdot%2024%5Ccdot%2025%20%3D%20124500-1200%20%3D%20123.300)