I'd like to find the last three digits of
$$S= \sum_{i=2}^{10^7}\bigl(i^7 + i^5\bigr)$$
I tried to calculate the sum by using partial sums and by process of similarity and periodicity of last digits but I can't proceed with that. I have found that the last digit is $0$ but I can't find the last three digits. Please help me to do it.
The three last digits of $i^7+i^5$ repeat every 1000 numbers, so if the lower limit of the sum were $i=1$, then you would have $10\,000$ repetitions of the same thousand terms modulo $1000$. This gives $10000x$ for some $x$, and the last three digits of that are obviously $000$.
Can you now correct for the fact that the term $1^7+1^5$ is not there?