X_(n+1)-aX_n+2=0 Is a linear non homogeneous difference equation.
It's solution can be composed of a solution for the homogeneous
X_n^h such that X_(n+1)^h-aX_n^h=0
plus a particular solution X_n^p for the non homogeneous equation such that
X_(n+1)^p-aX_n^p+2=0
For the homogeneous solution the proposal is
X_n^h=Ca^n. Substituting
Ca^(n+1)-aCa^n= Ca^(n+1)-C a^(n+1)=0
Now for the particular we propose
X_n^p=C_na^n then
C_(n+1)a^(n+1)-aC_na^n=a^(n+1)(C_(n+1)-C_n)=2
so
C_(n+1)-C_n=2/a^(n+1)
then beginning with C_0 we have
C_1= C_0 + 2/a
C_2=C_1+2/a^2 = C_0 +2/a+2/a^2
...
and then
C_n = C_0+2 sum_(k=1)^na^(-k) and finally
X_n = X_n^h+X_n^p=(C_0+2 sum_(k=1)^na^(-k))a^n=C_0a^n+2sum_(k=1)^na^k
If we need that X_n < X_(n-1) then
C_0a^n+2sum_(k=1)^na^k < C_0a^(n-1)+2sum_(k=1)^(n-1)a^k
or
C_0a^n+2a^n < C_0a^(n-1) or
a < C_0/(C_0+2)
Here C_0 is the initial condition.
