Alternating sum of sequence

Let $S_n = x_1 - x_2 + x_3 - \dots + (-1)^{n+1} x_n = n^2 + 2n$.

For $n = 48$: $S_{48} = x_1 - x_2 + \dots - x_{48} = 48^2 + 2(48)$

For $n = 49$: $S_{49} = S_{48} + x_{49} = 49^2 + 2(49) \implies x_{49} = 49^2 + 2(49) - (48^2 + 2(48)) = (49^2 - 48^2) + 2 = 97 + 2 = 99$

For $n = 50$: $S_{50} = S_{49} - x_{50} = 50^2 + 2(50) \implies x_{50} = S_{49} - (50^2 + 2(50)) = 49^2 + 2(49) - (50^2 + 2(50)) = -(50^2 - 49^2) - 2 = -99 - 2 = -101$

Sum $x_{49} + x_{50} = 99 + (-101) = -2$.