Geometric sequences

The sum of the first three terms of a geometric sequence is \( \frac78 \) and the common ratio is \( \frac12 \). Find the sum of all the terms from the \( 3 \)rd to the \( 5 \)th of the sequence.

The first term of a geometric sequence is \( 3 \). Find the common ratio so that the sum of the first \( 3 \) terms is minimal.

Find the second term and the common ratio of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} a_2-a_1&=22, \\ a_3-a_2&=66. \end{aligned} \]

Find the sum of the second and the third term of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} \frac{a_4}{a_1}&=-8, \\ a_4-a_2&=-9. \end{aligned} \]