I encounter a problem about genetics equilibrium under the condition of mutation that needs me to ask for whether the following sequence converges to a point and if yes, what that point is.
$$\eqalign{ & {a_1} = 0.5 \cr & u = 0.7 \cr & v = 0.55 \cr} $$
$$\eqalign{ & {a_1} \in (0,1) \cr & {a_2} = {a_1}(1 - u) + v(1 - {a_1}){\rm{ }} \cr & u,v \in (0,1) \cr & {a_3} = {a_2}(1 - u) + v(1 - {a_2}) \cr & {a_n} \to ?{\rm{ }} \cr} $$
when $$n \to \infty $$
The more important question is how is the answer derived? As a biomedical sciences student I only have background in Calculus I, II and linear algebra. No analysis background.
We can rewrite your equations as: $$a_2=a_1(1-u-v)+v$$ $$a_3=a_2(1-u-v)+v$$ and more generally as: $$a_{n+1}=a_n(1-u-v)+v \tag{2}$$ the idea to solve such a relation is to first find a constant solution $c \in \Bbb R$: $$c=(1-u-v) c+v \tag{2}$$ so: $$c=\frac{v}{u+v}$$ Then by taking $(1)-(2)$: $$(a_{n+1}-c)=(a_n-c)(1-u-v)$$ so $(a_n-c)$ is a geometric progression. Here as $|1-u-v| <1$ you have: $$\lim_{n \to \infty} (a_n-c)=0$$ so: $$\lim_{n \to \infty} a_n=c=\frac{v}{u+v}$$