31. a) The third term of an arithmetic progression is equal to 0. Find the sum of the first 5 terms.

b) The third term of a geometric progression is equal to 4. Find the product of the first 5 terms.
31. a) We know that for an arithmetic progression $u_{1}=u_{3}-2 d ; u_{2}=u_{3}-d$; $u_{4}=u_{3}+d ; \quad u_{5}=u_{3}+2 d$. Therefore, the sum of the first five terms is

$$
\begin{aligned}
& \delta_{5}=u_{1}+u_{2}+u_{3}+u_{4}+u_{5}=\left(u_{3}-2 d\right)+ \\
& +\left(u_{3}-d\right)+u_{3}+\left(u_{3}+d\right)+ \\
& +\left(u_{3}+2 d\right)=5 u_{3}=0
\end{aligned}
$$

b) We need to find

$$
P_{5}=u_{1} \cdot u_{2} \cdot u_{3} \cdot u_{4} \cdot u_{5}
$$

We know that for a geometric progression

$\boldsymbol{u}_{1}=\frac{u_{3}}{q^{2}} ; \quad u_{2}=\frac{u_{3}}{q} ; \quad u_{4}=u_{3} q ; \quad u_{5}=u_{3} q^{2}$.

Therefore,

$$
P_{5}=\frac{u_{3}}{q^{2}} \cdot \frac{u_{3}}{q} \cdot u_{3} \cdot u_{3} q \cdot u_{3} q^{2}=u_{3}^{5}=4^{5}=1024
$$