The terminal wealth is the result of a product of independent, equally distributed random variables e.g.

Final Wealth = (R1*R2*R3* ... *R1000)*Initial Wealth

where R is your discrete process with mean 1.25 and variance 1.125.

Therefore, unless I'm mistaken, it will follow a log-normal distribution (take the logs of both sides in the expression above - the sum of logs will be approximately normal if we appeal to the central limit theorem). If you look at a histogram of the log of the terminal wealth, this might be more instructive. You can do this in R doing something like:

terminal.wealth = numeric(1000000)
for(i in 1:1000000){
# Each example has 1000 coin tosses
s = sample(c(2,0.5),1000,replace=TRUE)
s = c(1,cumprod(s))
terminal.wealth[i] = tail(s,1)
}
plot(density(log(terminal.wealth)),lwd=3)
lines(seq(-100,100,by=0.01),dnorm(seq(-100,100,by=0.01),
mean=mean(log(terminal.wealth)),sd=sd(log(terminal.wealth))),
col="red",lwd=3)

The mean of the log terminal wealth is about zero, hence the median of the log normal variable (the terminal wealth) is exp(0) = 1 i.e. you gain nothing. In the above example with a 1000 coin tosses, the variance of the log terminal wealth is about 480, so the mean of the terminal wealth itself is exp(480/2) = 1.7e+104.

The wiki link to the log normal distribution might also be helpful:
https://en.wikipedia.org/wiki/Log-normal_distribution