Explain how the common ratio of a geometric sequence can be found.
Find the common ratio of the following geometric sequences:
2, - 16, 128, - 1024, \ldots
- 9, 8.1, - 7.29, 6.561,\ldots
- 70.4, - 17.6, - 4.4, - 1.1, \ldots
Write down the next two terms for the following sequences:
4, 12, 36, \ldots
12, -48, 192, \ldots
1, \dfrac {3}{4}, \dfrac {9}{16}, \ldots
- 6, 9, - \dfrac {27}{2}, \ldots
Write down the next three terms for the following sequences:
Consider the first four terms in this geometric sequence: - 8, - 16, - 32, - 64,\ldots
Evaluate:
\dfrac {T_2}{T_1}
\dfrac {T_3}{T_2}
\dfrac {T_4}{T_3}
Hence, find the value of T_5.
For each of the following, write the first four terms in the geometric progression:
The first term is 6 and the common ratio is 4.
The first term is 7 and the common ratio is - 2.
The first term is 700\,000 and the common ratio is 1.04.
The first term is - 2 and the common ratio is 3.
The first term is 1.3 and the common ratio is - 4.
Consider the sequence: 4, - 28, 224, - 1372, \ldots
Is the sequence arithmetic, geometric or neither?
For each of the following:
Determine if the sequence is arithmetic or geometric. Explain your answer.
Find the common ratio or common difference, whichever is applicable.
11, - 99, 891,- 8019, \ldots
2, 6, 10, 14, \ldots
Find the missing terms in the following geometric progressions:
- 5, x, - 80, 320, y
a, b, \dfrac {3}{25}, - \dfrac {3}{125}, c
State whether each of the following sequences is a geometric progression:
4, - 4, 4, - 4, \ldots
1, \sqrt{6}, 6, 6 \sqrt{6},\ldots
2, 0, - 2, - 4,\ldots
2, 2^{2}, 2^{4}, 2^{6}, \ldots
Suppose t_{1}, t_{2}, t_{3}, t_{4}, t_{5},\ldots is a geometric sequence.
Is t_{1}, t_{3}, t_{5},\ldots also a geometric sequence? Explain your answer.
For each of the given explicit rules:
List the first four terms of the sequence.
State the common ratio.
T_n = 3 \times 4^{n - 1}
T_n = - 4 \times \left( - 3 \right)^{n - 1}
For the following geometric sequences:
State the general or explicit rule for the nth term of the sequence.
Find T_{10}.
If the first term of a sequence is 27 and the common ratio is \dfrac {1}{3}, find the 10th term.
If the first term of a sequence is 1.2 and the common ratio is - 5, find the 4th term.
If the first term of a sequence is 400\,000 and the common ratio is 1.12, find the 3rd term.
In a geometric progression, T_7 = \dfrac {64}{81} and T_8 = \dfrac {128}{243}.
Find the value of r, the common ratio in the sequence.
Find the first three terms of the geometric progression.
In a geometric progression, T_4 = 32 and T_6 = 128.
Find the value of r, the common ratio in the sequence.
For the case where r is postive, find the value of a, the first term in the sequence.
Consider the sequence in which the first term is positive.
Find an expression for T_n, the nth term of this sequence.
Find three positive values between 18 and \dfrac {32}{9} such that the five terms form successive terms in a geometric progression.
Find three consecutive positive terms of a geometric progression if they have a product of 125 and the third term is 9 times the first.
Consider the finite sequence: 4, 4 \sqrt{2}, 8, \ldots, 256
Find the common ratio.
Find T_6.
Solve for n, the number of terms in the sequence.
Find the common ratio for the geometric sequence where the first two terms are \sqrt{5} + \sqrt{3} and \sqrt{5} - \sqrt{3}.
Consider the following:
1, x and y are the first three terms of an arithmetic sequence. Form an equation for y in terms of x.
1, y and x are also the first three terms in a geometric sequence. Form an equation for x in terms of y.
Hence, solve for the values of y.
One solution is y = 1 and x = 1 which produces the sequence 1, 1, 1.
Find the first three values of the arithmetic sequence for the other solution for x and y, along with the common difference.
Find the first three values of the geometric sequence for the other solution for x and y, along with the common ratio.
Consider the recurrence relation: u_{n + 1} = 2 u_n, u_1=4.
Complete the table of values:
Find u_7.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| u_n | 4 |
The nth term of a geometric progression is given by the equation T_n = 25 \times \left(\dfrac {1}{5}\right)^{n - 1}.
Complete the table of values:
Find the common ratio.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
The nth term of a geometric progression is given by the equation T_n = - 72 \times \left( - \dfrac {4}{3} \right)^{n - 1}.
Complete the table of values:
Find the common ratio.
| n | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| T_n |
Each of the given tables of values represents terms in a geometric sequence:
Find r, the common ratio between consecutive terms.
Write a simplified expression for the nth term of the sequence, T_n.
Find the missing term in the table.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n | 5 | 40 | 320 | 2560 |
| n | 1 | 2 | 3 | 4 | 12 |
|---|---|---|---|---|---|
| T_n | 7 | -21 | 63 | -189 |
| n | 1 | 3 | 6 | 9 | 11 |
|---|---|---|---|---|---|
| T_n | -5 | -45 | -1215 | -32\,805 |
| n | 1 | 2 | 3 | 4 | 7 |
|---|---|---|---|---|---|
| T_n | -2 | -\dfrac {16}{3} | -\dfrac {128}{9} | -\dfrac {1024}{27} |
The values in the table show the nth term in a geometric sequence for consecutive values of n. Complete the missing values in the table:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | 5 | -320 |
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | -27 | -64 |
The given table of values represents terms in a geometric sequence:
Find r, the common ratio between consecutive terms.
Write a simplified expression for the nth term of the sequence, T_n.
| n | 1 | 4 | 9 |
|---|---|---|---|
| T_n | -9 | 576 | -589\,824 |
The nth term of a geometric progression is given by the equation T_n = 2 \times 3^{n - 1}
Complete the table of values:
Find the common ratio between consecutive terms.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
Plot the points in the table that correspond to n = 1, 2, 3 and 4 on a cartesian plane.
State whether the joined points would form a straight line or a curve.
The nth term of a geometric progression is given by the equation T_n = 6 \times \left( - 2 \right)^{n - 1}
Complete the table of values:
Find the common ratio between consecutive terms.
| n | 1 | 2 | 3 | 4 | 11 |
|---|---|---|---|---|---|
| T_n |
Plot the points that correspond to n = 1, 2, 3 and 4 on a cartesian plane.
Describe the shape of the joined points.
The plotted points represent terms in a geometric sequence:
Find the first term in the sequence.
Identify r, the common ratio between consecutive terms.
Write a simplified expression for the nth term of the sequence, T_n.
The plotted points represent terms in a geometric sequence:
Complete the table of values for the given points:
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n |
Identify r, the common ratio between consecutive terms.
Write a simplified expression for the nth term of the sequence, T_n.
Find the 10th term of the sequence.
The plotted points represent terms in a geometric sequence:
Complete the table of values for the given points:
| n | 1 | 2 | 3 |
|---|---|---|---|
| T_n |
Identify r, the common ratio between consecutive terms. Assume all values in the series are negative.
Write a simplified expression for the nth term of the sequence, T_n.
The plotted points represent terms in a geometric sequence:
Complete the table of values for the given points.
| n | 1 | 2 | 3 |
|---|---|---|---|
| T_n |
Identify r, the common ratio between consecutive terms.
Find the 4th term of the sequence.
Write a simplified expression for the nth term of the sequence, T_n.
Find the 10th term of the sequence.
The plotted points represent terms in a geometric sequence:
Identify r, the common ratio between consecutive terms.
Write a simplified expression for the nth term of the sequence, T_n.
The points are reflected about the horizontal axis to form three new points.
If these new points represent consecutive terms of a geometric sequence, write the equation for T_k, the kth term in this new sequence.