Calculate the value of: \[1\cdot \left(1+\frac{1}{2025} \right)^1 + 2\cdot \left(1+\frac{1}{2025} \right)^2 +\dots + 2025\cdot \left(1+\frac{1}{2025} \right)^{2025},\] and provide proof that your calculation is correct.