Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.
Source: Slovenia National MO 2005 4th Grade P2
Tags: Sequence, algebra
Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.