How can one convert a recursive sequence into an explicit one given the following? $$a_1=0 $$ $$~a_{n+1} = 0.5a_n + 1$$
Looking at the values of the sequence and their differences $$ 0,1,1.5,1.75,1.875 ... $$ $$1,0.5,0.25,0.125...$$
I still wasn't able to come up with a function describing this relationship. What is the approach one does in such cases?
Any help is very appreciated, thank you.
hint
we have $$(\forall n,k \ge 2)\;2^ka_n=2^{k-1}a_{n-1}+2^k$$ So
$$2a_2=a_1+2$$ $$2^2a_3=2a_2+2^2$$ $$2^3a_4=2^2a_3+2^3$$
$$\cdots$$ $$2^{n-1}a_n=2^{n-2}a_{n-1}+2^{n-1}$$
the sum gives $$2^{n-1}a_n=a_1+2+2^2+2^3+\cdots+2^{n-1}$$