Problem 85

Solution:

$(1)$ Let the common difference of the arithmetic sequence $\{a_n\}$ be $d$, then $a_3+a_8-(a_2+a_7)=2d=-6$, $\therefore d=-3$,

$\therefore a_2+a_7=2a_1+7d=-23$, solving this gives $a_1=-1$, $\therefore$ the general formula for the sequence $\{a_n\}$ is $a_n=-3n+2$;

$(2)$ $S_n = n \times (-1) + \frac{n(n-1)}{2} \times (-3) = -\frac{3}{2}n^2 + \frac{3}{2}n$.

Therefore, the answers are:

$(1)$ The general formula for the sequence is $\boxed{a_n=-3n+2}$.

$(2)$ The sum of the first $n$ terms of the sequence $\{a_n+1\}$ is $\boxed{S_n = -\frac{3}{2}n^2 + \frac{3}{2}n}$.