1003112802
Level:
A
The first term of a geometric sequence is \( -36 \) and the fourth term is \( -\frac{32}3 \). Find the common ratio.
\( \frac23 \)
\( -\frac23 \)
\( \frac32 \)
\( -\frac32 \)
\( \frac13 \)
Geometric sequences
1003112801
Level:
A
The third term of a geometric sequence is \( 100 \) and the common ratio is \( -5 \). Find the first term of this sequence.
\( 4 \)
\( -4 \)
\( 20 \)
\( -20 \)
\( 500 \)
1003134709
Level:
B
Find the sum of the second and the third term of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} a_1-a_2&=b, \\ a_1+a_2&=-3b, b\in\mathbb{R}. \end{aligned}\]
\( -6b \)
\( -5b \)
\( -3b \)
\( -2b \)
\( 2b \)
1003134708
Level:
B
Find the sum of the second and the third term of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} \frac{a_4}{a_1}&=-8, \\ a_4-a_2&=-9. \end{aligned} \]
\( 3 \)
\( -3 \)
\( \frac32 \)
\( -2 \)
\( \frac92 \)
1003134707
Level:
B
Find the sum of the first five terms of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} a_1+a_3&=8, \\ a_1+a_2&=0. \end{aligned} \]
\( 4 \)
\( -4 \)
\( 0 \)
\( 8 \)
\( -8 \)
1003134706
Level:
B
Find the second term and the common ratio of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[ \begin{aligned} a_2-a_1&=22, \\ a_3-a_2&=66. \end{aligned} \]
\( a_2=33 \), \( q=3 \)
\( a_2=11 \), \( q=3 \)
\( a_2=22 \), \( q=3 \)
\( a_2=33 \), \( q=2 \)
\( a_2=11 \), \( q=2 \)
1003134705
Level:
B
Find the first term of the geometric sequence \( \{a_n\}_{n=1}^{\infty} \), if: \[\begin{aligned} a_3\cdot a_4&=-25, \\ a_4-a_3&=10. \end{aligned} \]
\( -5 \)
\( 5 \)
\( -1 \)
\( 1 \)
\( -15 \)
1003134704
Level:
B
The first term of a geometric sequence is \( 3 \). Find the common ratio so that the sum of the first \( 3 \) terms is minimal.
\( -\frac12 \)
\( -5 \)
\( 0 \)
\( 2 \)
\( -1 \)
1003134703
Level:
B
The first term of a geometric sequence is different from zero and \( s_3=-3s_2 \). (\( s_n \) is the sum of the first n terms.) Find the common ratio.
\( -2 \)
\( 2 \)
\( \frac12 \)
\( 0 \)
\( -3 \)
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.
\( \frac7{32} \)
\( \frac{31}{32} \)
\( \frac3{32} \)
\( \frac78 \)
\( \frac58 \)