Minimal Number of Symbols in Context-Free Grammar for a Special One-Letter Language Given is the language
$$L_n = \{a^n\},$$
where $n$ is a natural number and $a$ is a letter. What are the productions/rules of a minimal context-free grammar according to the number of symbols of the productions (e.g. the number of symbols of the productions $\{ S \rightarrow a, S \rightarrow aS \}$ is 7).

For $n = 1$ the (only) rule is $\{ S \rightarrow a \}$ and for $n = 2$ we get $\{ S \rightarrow aa \}$, which can be proved easily. The number of symbols are $3$ and $4$, respectively. You may think, that $\{ S \rightarrow a^n \}$ ($2 + n$ Symbols) is always the shortest rule according the number of symbols for $L_n$ but $L_{10} = \{a^{10}\}$ can be constructed by $\{ S \rightarrow A^5, A \rightarrow aa \}$ that has only $11$ symbols (instead of $2 + 10 = 12$). This looks like an optimal compression problem.
This is essentially the problem of "addition chains", discussed in great detail in many papers and in Knuth's The Art of Computer Programming.  In brief:  the problem as you have stated it is not known to be NP-hard, but nobody knows an efficient algorithm to find the absolutely minimum.  However, there are good algorithms that come reasonably close to the minimum.