1. Let $a_{m+n}=A$ and $a_{m-n}=B$ be terms of an arithmetic progression. Express the terms $a_{n}$ and $a_{m}$ in terms of $A$ and $B$.
Solution. We have

$$
a_{m+n}=a_{1}+(m+n-1) d=A
$$

and

$$
a_{m-n}=a_{1}+(m-n-1) d=B
$$

which gives

$$
a_{m}+n d=A \text { and } a_{m}-n d=B,
$$

hence $d=\frac{A-B}{2 n}$. Now from $a_{m}-n d=B$ and $a_{n}+m d=A$ it follows that

$$
a_{m}=B+n d=B+n \frac{A-B}{2 n}=\frac{A+B}{2} \text { and } a_{n}=A-m d=A-m \frac{A-B}{2 n}=\frac{(2 n-m) A+m B}{2 n}
$$