  In this paper we study the theory of addition chains producing any given
number $n\geq 3$. With the goal of estimating the partial sums of an additive
chain, we introduce the notion of the determiners and the regulators of an
addition chain and prove the following identities \begin{align}\sum
\limits_{j=2}^{\delta(n)+1}s_j=2(n-1)+(\delta(n)-1)+a_{\delta(n)}-r_{\delta(n)+1}+\int
\limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}r_jdt.\nonumber
\end{align}where \begin{align}
2,s_3=a_3+r_3,\ldots,s_{\delta(n)}=a_{\delta(n)}+r_{\delta(n)},s_{\delta(n)+1}=a_{\delta(n)+1}+r_{\delta(n)+1}=n\nonumber
\end{align}are the associated generators of the chain
$1,2,\ldots,s_{\delta(n)},s_{\delta(n)+1}=n$ of length $\delta(n)$. Also we
obtain the identity\begin{align}\sum
\limits_{j=2}^{\delta(n)+1}a_j=(n-1)+(\delta(n)-1)+a_{\delta(n)}-r_{\delta(n)+1}+\int
\limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}r_jdt.\nonumber
\end{align}
