If $\displaystyle p_{n}$ 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 $\displaystyle p_{n}$. and show that the general solution is $\displaystyle p_{n}=A(3)^n+B$

my answer for i)

let $\displaystyle G_{n}$ represent the growth of the insect in year n; $\displaystyle P_{n+1}=P_{n}+G_{n}$

ii)$\displaystyle P_{n-1}=P_{n-2}+G_{n-2}$

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