So I'm trying to learn difference equations, and the book that I'm using defines the following:
The discrete derivative of a function $a_n$ of the integers is defined as:
$$ D a_n = a_{n+1} - a_n $$
Similarly the discrete antiderivative or integral $b_n$ of the function $ a_n $ is defined as the sum
$$ b_n = \sum_{ j = n_0 }^n a_ j $$
My first question is that, n this sense does antiderivate and derivative are inverse operation, because
$$ \sum_{j = n_0 }^n D a_n = a_{n + 1} - a_n \neq a_n $$
and also when I'm solving the following differential equation
$$ a_{n+1} = p(n)a_n $$
The algebraic way to solve it says that, take logarithms and define $b_n = \ln a_n $ so that
$$ b_{n + 1} - b_n = \ln p(n) $$
and the left hand side is an exact discrete derivative, and the solution is obtained by computing the discrete integral, and somehow this happends
$$ b_n = b_1 + \sum_{j = 1}^{n - 1} \ln p(j) $$
my second question is how, and why? Since the definition starts from $ n_0 $ and I do not know how is tat we compute from $ 1 $ to $ n - 1 $ and the existence of $ b_1 $. Thanks.