# exponential growth calculated in two ways

Maybe quite basic question, but was little surprise for me.

Lets say we start with $2$ units (maybe thousands of microbes) and we have $30 \%$ increase (growth rate) over time unit. The question is how large their population will be after $12$ time units.

I think this formula will apply: $$2 \cdot \exp(0.3 \cdot 12) \approx 73$$ But I had some doubt in so large result so I did "long way": $$\begin{array}{ccccc} 2 &+& 2 \cdot 0.3 &=& 2.6 \\ 2.6 &+& 2.6 \cdot 0.3 &=& 3.38 \\ &&\ldots \end{array}$$ and after $12$ steps I get $\approx 47$ which is approximately what I expected. So how can I calculate this in short way? How that first calculation is wrong, how different it is?

A 30% increase means your quantity is multiplied by $1.3$ each time. So the right formula is: $$1.3^{\ t}\quad\hbox{or, if you prefer:}\quad e^{\ t\ln1.3}.$$ Substitute here $t=12$.