Efficient approximation of nth term without losing accuracy

Question

Probelem Given a recurrence relation for g_n as

g₀ = c where is a contant double.
g_n = f( g_n-1 ) , where f is a linear function

then find the value of another recurrence given by

h_n = g_n/exp(n)

constraints: 1 <= n <= 10^9

Mathematically, the value of g(n) can be calculated in log(n) time and then h(n) can be calculated very easily but the problem is the overflow of doubles data types. So the above strategy works only for n around 1000 but not for bigger n. Note that value of h(n) can be well within the range of doubles

The actual problem is that we are trying to calculate h(n) from g(n). My question is that is there any good method to caculate h(n) directly without overflowing doubles.

Answer 1

G0=c
G1=ac+b
G2=a²c+ab+b
G3=a³c+a²b+ab+b
...
Gn=a^nc+b(a^n-1)/(a-1)

Then

Hn = (a/e)^nc+b((a/e)^n-1/e^n)/(a-1) ~ (a/e)^n (c + b/(a-1))