Neighbours of a midpoint level

Write both numbers as $50$ plus or minus something.

1. Write $49=50-1$ and $51=50+1$.
2. The cross terms $\pm 2\times 50$ cancel in the sum.
3. So $49^2+51^2=2\times 50^2+2\times 1^2$.
4. This is $5000+2=5002$.

Answer: $5002$