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. 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 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 |