Chapter 12 Sequences and Series - 12.2 Analyze Arithmetic Sequences and Series - 12.2 Exercises - Skill Practice - Page 807: 57

$n=25$

For an arithmetic series, the sum for the finite series is given by: $S_n=\dfrac{n(a_1+a_n)}{2}$ We are given that $S_n=-1150$ Now, $-1150=\dfrac{n \times (50+58-8n)}{2}$ or, $2n^2-27n-575=0$ or, $(n-25) (2n+23)=0$ This gives: $n=25, \dfrac{-23}{2}$ Hence, the positive solution for $n$ is $n=25$