jmc

$a_n$ is positive when $n$ is odd and negative when $n$ is even. Consider an arbitrary odd number $j.$ It follows that $$a_j+a_{j+1}=(3j+2)-(3(j+1)+2)=-3.$$ In $a_1+a_2+\cdots +a_{100},$ there are $50$ such pairings, so the sum is $(-3)(50)=\boxed{-150}.$