State the first four terms in the sequences defined by the following recursive rules:
T_{n+1} = 4 T_n,\text{ } T_1 = 0.5
T_{n + 1} = 0.2 T_n,\text{ } T_1 = 25
T_n = 2 T_{n - 1},\text{ } T_5 = 32
T_n = \dfrac{1}{4} T_{n - 1},\text{ } T_5 = \dfrac{1}{4}
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
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.
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.
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}
Consider the following recursive definition which starts from T_1: T_n = 2 T_{n - 1},\ T_5 = 32
State the first 4 terms in the sequence.
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.
Consider the following sequence: 20, 40, 80, 160, \ldots
By finding the rule for the nth term, find the value of n corresponding to the term 640.
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.
Find the three consecutive terms.
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 |
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 |
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 |