The way these recurrence relations are taught at university is to distinguish the new value from the old value by using a ' pronounced prime.

Here, you would write x' = x + 1 to give the recurrence relation x[n+1] = x[n] + 1. Or, more generally x' = f(x) for x[n+1] = f(x[n]). The reason for this is because in mathematics x = x + 1 is absurd (ignoring modulo arithmetic).