 Pairing proof for an arithmetic series

Reverse the finite list of sequence terms and add it to the original list. At position $k$,Every position therefore contributes the same value. Reversal preserves the sum, giving $2S_n=n(a_1+a_n)$ and the stated formula. The symmetry explains why the endpoint average represents the whole list; one does not need to add each entry separately.

The reversal argument works for both even and odd lengths. For $2,5,8,11,14$, the two outer pairs each total sixteen and the middle entry is eight. Adding the list to its reverse counts that middle entry twice too, giving $2S_5=5\times 16$ and $S_5=40$. No half-entry is required for an odd number of entries.