Basic Rhythms

All the results can be checked by calculation. For an explanation and proof it is most efficient to use a general method that works for all bases. Moreover algebra reveals the structure that is obscured when using particular numbers to check special cases. Writing $b$ for the base then each expression can be written in terms of $b$ and, by collecting like terms in the expression, or equating the powers of $b$, it can be shown that each expression holds. All that is required is some care over the algebraic manipulation.