#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.
