Chapter 12 - Sequences; Induction; the Binomial Theorem - Chapter Review - Review Exercises - Page 839: 20

The common difference is: $d=5.$ The initial term: $a_1=1.$ $a_n=-4+5n$, $a_n=a_{n-1}+5$

We know that $a_{7}=31,a_{20}=96$. Thus the common difference is: $d=\frac{a_k-a_l}{k-l}=\frac{a_{20}-a_{7}}{20-7}=\frac{96-31}{13}=5.$ The initial term: $a_1=a_n-(n-1)d=a_7-(6)d=31-(6)5=1.$ Thus: $a_n=a_1+(n-1)d=1+(n-1)5=-4+5n$, $a_n=a_{n-1}+d=a_{n-1}+5$