Chapter 14 - Sequences, Series, and the Binomial Theorem - 14.2 Arithmetic Sequences and Series - 14.2 Exercise Set - Page 902: 31

$a_1=1$ $d=1$

We have an arithmetic sequence in which $a_{13}=13$ and $a_{54}=54.$ In order to find $a_{1}$ and $d$ we will use the fact that any term of arithmetic sequence is expressed through the formula $a_{n}=a_1+d(n-1)$ Hence, $a_{13}=a_1+d\cdot12$ $a_{54}=a_1+d\cdot53$ $a_{54}-a_{13}=a_1+d\cdot53-a_1-d\cdot12=d(53-12)=41d$ $54-13=41d$ So $d=1$ Nowe we can use any equation to find $a_1.$ $13=a_1+1\cdot12$ $a_1=1$