Geometric sequences

1003134706

Level: 
B
Find the second term and the common ratio of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} a_2-a_1&=22, \\ a_3-a_2&=66. \end{aligned} \]
\( a_2=33 \), \( q=3 \)
\( a_2=11 \), \( q=3 \)
\( a_2=22 \), \( q=3 \)
\( a_2=33 \), \( q=2 \)
\( a_2=11 \), \( q=2 \)

1003134702

Level: 
B
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.
\( \frac7{32} \)
\( \frac{31}{32} \)
\( \frac3{32} \)
\( \frac78 \)
\( \frac58 \)

1003084910

Level: 
A
We were given a geometric sequence \( \frac12\text{, }\ \frac14\text{, }\ \dots \). What is the formula for the \( n \)th element of the sequence?
\( a_n=\frac1{2^n}\text{, }\ n\in\mathbb{N} \)
\( a_n=\frac1{2^{n+1}}\text{, }\ n\in\mathbb{N} \)
\( a_n=\frac1{2^{n-1}}\text{, }\ n\in\mathbb{N} \)
\( a_n=\frac1{2^{2n}}\text{, }\ n\in\mathbb{N} \)