Chapter 14 - Sequences, Series, and the Binomial Theorem - 14.2 Arithmetic Sequences and Series - 14.2 Exercise Set - Page 903: 61

See proof

If $p$, $m$ and $q$ are consecutive terms in an arithmetic sequence then $p,$ $m=p+d,$ $q=m+d=p+2d,$ where d is the common difference between consecutive terms. Then $p+q=p+p+2d=2p+2d=2m$ Therefore $m=\frac{p+q}{2}$