Chapter 14 - Sequences, Series, and the Binomial Theorem - 14.2 Arithmetic Sequences and Series - 14.2 Exercise Set - Page 902: 22

$n=46$

The n th term can be calculated as: $a_n=a_1+d\times(n-1)$ Here: $a_n=-27$ and we have to find $n$. In an arithmetic sequence, first we have to find the common difference, which can be calculated as the difference of any two subsequent terms: $d=a_2-a_1=\frac{7}{3}-3=-\frac{2}{3}$. By substituting we get: $-27=3+(-\frac{2}{3})\times(n-1)$ $n-1=\frac{-27-3}{-\frac{2}{3}}=45$ $n=46$