$$\text{Let } a_1, a_2, \dots \text{ be integers such that } a_1 - a_2 + a_3 - a_4 + \dots + (-1)^{n-1} a_n = n, \quad \text{for } n \geq 1.$$ $$\text{Then } a_{51} + a_{52} + \dots + a_{1023} \text{ equals ?}$$