For each of the following geometric sequences, evaluate:
\dfrac{T_2}{T_1}
\dfrac{T_3}{T_2}
\dfrac{T_4}{T_3}
T_5
- 9 , - 10.8 , - 12.96 , - 15.552 , \ldots
- 8, -16, -32, -64, \ldots
Write down the next two terms in the following sequences:
4, 12, 36, \ldots
12, -48, 192, \ldots
- 6, 9, - \dfrac{27}{2}, \ldots
Consider the sequence -1, -7, -49, \ldots
Find the next term of the sequence.
Find the 5th term of the sequence.
Find the 6th term of the sequence.
Determine the common ratio of the following geometric sequences:
2, - 16, 128, - 1024, ...
- 70.4,- 17.6,- 4.4,- 1.1,...
Write the first 5 terms of the following sequences given the first term and the common ratio:
First term: - 2, common ratio: 3
First term: 1.3, common ratio: - 4.
First term: 700\,000, common ratio: 1.04.
The first term of a sequence is 3.9 and the common ratio is 2.
State the 5th term.
Calculate the sum of the first 5 terms.
Find the missing terms in the geometric sequence below:
x, \dfrac{3}{25}, - \dfrac{3}{125}, y, \ldots
If t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, \text{. . .} is a geometric sequence, is t_{1}, t_{3}, t_{5}, \text{. . .} a geometric sequence?
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.
List the first three terms of the geometric progression.
For each of the following sequences, write the recursive rule for T_n in terms of T_{n-1} including the initial term T_1:
9, 45, 225, 1125, \ldots
2, 20, 200, 2000, \ldots
5, - 15 , 45, - 135 , \ldots
List the first 5 terms of the following sequences defined by:
a_1 = 2 and a_{n+1} = 3 a_n
a_1 = 8 and a_n = \dfrac{1}{2} a_{n - 1}
The first term of a geometric sequence is 7, and the fourth term is 875.
Find the common ratio, r, of this sequence.
Write the recursive rule for T_{n+1} in terms of T_n, including the initial term T_1.
Calculate the sum of the terms between the 4th and 9th term inclusive.
The first term of a geometric sequence is 6, and the third term is 96.
Find the possible values of the common ratio, r, of this sequence.
Write the recursive rule for T_n in terms of T_{n - 1} that uses the positive common ratio, and the initial term T_1.
Write the recursive rule for S_n in terms of S_{n - 1} that uses the positive common ratio, and the initial term S_1.
Determine the sum of the terms between T_4 and T_8 inclusive.
The fourth term of a geometric sequence is 16, and the seventh term is 128.
Find the common ratio, r, of this sequence.
Find the first term of this sequence.
Write the recursive rule for T_{n+1} in terms of T_n, including the initial term T_1.
Determine the sum of the terms between the 3rd and 10th term inclusive.
The third term of a geometric sequence is 7500, and the seventh term is 12.
Find the common ratio, r, of this sequence.
Find the first term of this sequence.
Write the recursive rule for T_n in terms of T_{n - 1} that uses the positive common ratio, and the initial term T_1.
Write the recursive rule for S_n in terms of S_{n - 1} that uses the positive common ratio, and the initial term S_1.
Calculate the sum of the first 15 terms of the sequence containing a negative common ratio. Round your answer to the nearest whole number.
Consider the following sequence:
3000, 600, 120, 24, \ldots
Write the recursive rule for T_{n+1} in terms of T_n , including the initial term T_1.
Determine the sum of the first 10 terms. Round your answer to the nearest whole number.
If the first term of a sequence is 90\,000 and the common ratio is 1.11:
Find the 6th term. Round your answer to three decimal places.
Calculate the sum of the first 6 terms. Round your answer to the nearest whole number.
Given the nth term of the following sequences defined by the given equation:
Write the first four terms of the sequence.
Find the common ratio.
T_n = 3 \times 4^{n - 1}
T_n = - 4 \times \left( - 3 \right)^{n - 1}
Consider the following sequence:
- 54 , - 18 , - 6 , - 2 , \ldots
Write the general rule for T_n, the nth term of the sequence.
Hence, state the next three terms of the sequence.
Find T_9, the 9th term of the sequence.
Calculate the sum of the first 9 terms. Round your answer to the nearest whole number.
If the first term of a sequence is 27 and the common ratio is \dfrac{1}{3}, find the 10th term.
Consider the following sequence:
- 0.3,- 1.5,- 7.5,- 37.5,...
Find the formula for the nth term of the sequence.
Hence, find the next three terms of the sequence.
In a geometric progression, T_4 = - 192 and T_7 = 12\,288.
Determine the value of r
Hence determine the value of a
Write an expression for T_n
In a geometric progression, T_4 = 192 and T_9 = 196\,608.
Find the value of r.
Hence determine the value of a.
Write an expression for T_n.
Insert three positive values between 18 and \dfrac{32}{9} such that the five terms form the successive terms in a geometric progression.
Three consecutive positive terms of a geometric progression have a product of 125. The third term is 9 times the first. Let the common ratio be r, and the middle term be b.
Find the the value of b.
Find the three consecutive terms.
State the value of r.
Consider the first-order recurrence relationship defined by T_n = 2 T_{n - 1},\text{ } T_1 = 2.
Determine the next three terms of the sequence from T_2 to T_4.
Plot the first four terms on a number plane.
Is the sequence arithmetic or geometric?
Consider the following sequence 5, - 10 , 20, - 40 , \ldots
Plot the first four terms on a cartesian plane.
Is the sequence arithmetic or geometric?
Consider the sequence plot drawn below:
State the terms of the first five points of the sequence.
Is the sequence arithmetic or geometric?
Write a recursive rule for T_{n+1} in terms of T_n and an initial condition for T_1.
Consider the sequence 54, 18, 6, 2, \ldots
If the sequence starts from n = 1, plot the first four terms on a number plane.
Is the relationship depicted by this graph linear, exponential or neither?
Write the recursive rule for T_{n+1} in terms of T_n, including the initial term T_1.
Find the sum of the first 10 terms. Round your answer to the nearest whole number.
Consider the sequence plot drawn below:
State the terms of the first five terms of the sequence.
Is the relationship depicted by this graph linear, exponential or neither?
Write the recursive rule for T_n in terms of T_{n - 1}, including the initial term T_1.
Calculate the sum of the first 12 terms. Round your answer to the nearest whole number.
Given the nth term of the following geometric progressions:
Complete the table of values.
Find the common ratio.
Plot the points in the table that correspond to n = 1, n = 2, n = 3 and n = 4 on a cartesian plane.
State whether the joined points would form a straight line, a wave shaped curve, a parabola, or an exponential curve.
T_n = 2 \times 3^{n - 1}.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
T_n = 6 \times \left( - 2 \right)^{n - 1}.
| n | 1 | 2 | 3 | 4 | 11 |
|---|---|---|---|---|---|
| T_n |
For the geometric progressions given by the equations below:
Complete the table of values.
Find the common ratio.
T_n = 25 \times \left(\dfrac{1}{5}\right)^{n - 1}
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
T_n = - 72 \times \left( - \dfrac{4}{3} \right)^{n - 1}.
| n | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| T_n |
For the geometric sequences given by the tables of values below:
Determine the common ratio, r.
Write an expression for the general nth term of the sequence, T_n.
Find the missing value 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 | 2 | 3 | 4 | 7 |
|---|---|---|---|---|---|
| T_n | -2 | -\dfrac{16}{3} | -\dfrac{128}{9} | -\dfrac{1024}{27} |
| n | 1 | 3 | 6 | 9 | 11 |
|---|---|---|---|---|---|
| T_n | -5 | -45 | -1215 | -32\,805 |
The given table of values represents terms in a geometric sequence:
Find r, the common ratio.
Write an expression for the general nth term of the sequence, T_n.
| n | 1 | 4 | 9 |
|---|---|---|---|
| T_n | -9 | 576 | -589\,824 |
For the following geometric sequences represented by the plotted points:
Complete a table of values for the given points.
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 10th term of the sequence.
The plotted points represent terms in a geometric sequence.
Complete a table of values for the points.
Find r, the common ratio. Assume all values in the series are negative.
Write an expression for the general nth term of the sequence, T_n.
Complete the missing values in the following tables that show the nth term in a geometric sequence:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | 5 | -320 |
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | -27 | -64 |
The plotted points represent terms in a geometric sequence:
State the first term in the sequence.
Find r, the common ratio.
Write an expression for the general nth term of the sequence, T_n.
The plotted points represent terms in a geometric sequence:
Find r, the common ratio.
Write an expression for the general 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.