Answer: [tex]P(x) = 50,000(1.12)^x[/tex]
Explanation: Let [tex]P(x)[/tex] be the population after x years. Since 50,000 is the initial population, P(0) = 50,000. Moreover, since the population increases 12% per year,
[tex]P(x) = P(x - 1) + 12\% (P(x - 1))
\\ = P(x - 1) + 0.12(P(x - 1))
\\ \boxed{P(x) = 1.12(P(x - 1))}[/tex]
Using the preceding equation:
[tex]P(1) = 1.12(P(0)) = 50,000(1.12)
\\ P(2) = 1.12(P(1)) = 1.12(50,000(1.12)) = 50,000(1.12)^2
\\ P(3) = 1.12(P(2)) = 1.12(50,000(1.12)^2) = 50,000(1.12)^3
\\.
\\.
\\.
\\P(x) = 1.12(P(x-1)) = 1.12(50,000(1.12)^{x-1}) = 50,000(1.12)^x
\\\boxed{P(x) = 50,000(1.12)^x}[/tex]
