If

denotes the population of a rapidly growing insect colony at the end of year n:

(i) write down an expression for the increase in population during year n;

(ii) write down an expression for the increase in population during year n-1;

(iii) given that the increase in population each year is three times the increase in the previous year formulate a second order differential equation for

. and show that the general solution is

my answer for i)

let

represent the growth of the insect in year n;

ii)

I'm wonder how to formulate a second order differential equation for this model? thanks