3. The number of different sequences $a_{1}, a_{2}, \cdots, a_{100}$ that satisfy $\left(a_{n}-1\right)\left(a_{n+1}+1\right)=0(n=1,2$, $\cdots, 99)$ is $\qquad$ .
3. Infinity.

For $\left(a_{1}-1\right)\left(a_{2}+1\right)=0$, we can take $a_{1}=1$, or $a_{2}=-1$, but it is not necessary to take $a_{1}=1$ and $a_{2}=-1$. Therefore, the sequence $a_{1}=1, a_{2}$ can be any value,
$$
a_{3}=a_{4}=\cdots=a_{100}=-1 \text {, }
$$
satisfies the condition, hence there are infinitely many such sequences.