R -forecast using a growth rate

Question

I have what I think to be a very simple question - I have a dataframe:

a <- c(1,2,3,4,5)
b <- (a/lag(a))-1

combining the two gives me a data frame:

df <- cbind(a,b)

      a         b
 [1,] 1        NA
 [2,] 2 1.0000000 
 [3,] 3 0.5000000
 [4,] 4 0.3333333
 [5,] 5 0.2500000

Now suppose I want to forecast (or put simply grow a) by the last growth rate in b. So for example, df$a[6] = df$a[5]*(1+df$b[5]), and df$a[7] = df$a[6]*(1+df$b[5]), up to n.

This kind of thing is obviously very easy to do in excel, but I can't seem to figure it out in R.

How do functions like predict() or forecast() work, especially when the variable is only autoregressive?