I originally solved this by using the "John von Neumann method" (explicitly summing the infinite series). But it's easier to solve recursively. We are looking for the value of K such that the variance of the EMA is equal to V/N (the variance of the ordinary N day moving average). From the formula for EMA, we have:


Var[EMA(t)] = K^2 Var[P(t)] + (1-K)^2 Var[EMA(t-1)]


The price variance is V and the variance of the ordinary N day moving average is V / N, so we want to solve for K in the equation:


V/N = K^2 V + (1-K)^2 (V/N)


Try solving this and see what you get.