How to prove recursive formula?

Question

A fish farmer has 4000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month.

(a) Show that the catfish population P_n after n months is given recursively by

P_n = 1.08P_n − 1 − 300    P₀ = 4000.

(b) How many catfish are in the pond after six months? (Round your answer to the nearest integer.)

Answer 1

The solution to this difference equation is P_n= 250(1.08)ⁿ + 3750 for n≥0.

The solution method will be found in "Calculus of Finite Differences" by Jordan

I will leave it to you to plug in n=6.

Alternatively, you can plug in the recursive relation to a spreadsheet.