xn+1 = xn + n - 1
x1 = -1
x2 = x1 + (1 - 1)
x3 = x2 + (2 - 1)
⇒ x3 = x1 + (1 - 1) + (2 - 1)
x4 = x3 + (3 - 1)
⇒ x4 = x1 + (1 - 1) + (2 - 1) + (3 - 1)
so, as we see that,
x100 = x1 + (1 - 1) + (2 - 1) + (3 - 1) + … + (99 - 1)
x100 = x1 + (1 + 2 + 3 + 4 + … + 98 + 99) - 99 (1)
x100 = x1 + (1 + 2 + 3 + 4 + … + 98)
x100 = (-1) + (\(98×\frac{99}{2}\))
x100 = (-1) + 4851
x100 = 4850