6.18 It is known that an arithmetic progression includes terms $a_{2 n}$ and $a_{2 m}$ such that $\frac{a_{2 n}}{a_{2 m}}=-1$. Does this progression have a term equal to zero? If yes, what is the index of this term?
6.18 We have

$$
\begin{aligned}
& \frac{a_{2 n}}{a_{2 m}}=-1 \Rightarrow a_{2 n}+a_{2 m}=0 \Rightarrow \\
& \Rightarrow\left[a_{1}+(2 n-1) d\right]+\left[a_{1}+(2 m-1) d\right]=0 \Rightarrow \\
& \Rightarrow a_{1}+(n+m-1) d=0
\end{aligned}
$$

This means that the $(n+m)$-th term of the progression is zero. Answer: yes; the $(n+m)$-th term.