Write down the next two terms for the following sequences:
4, 12, 36, \ldots
12, - 48, 192, \ldots
- 1, 8, - 64, \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.
Explain how the common ratio of a geometric sequence can be found.
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? Show working to justify your answer.
Consider the first four terms of the following geometric sequences:
Evaluate \dfrac{T_2}{T_1}.
Evaluate \dfrac{T_3}{T_2}.
Evaluate \dfrac{T_4}{T_3}.
Hence, find T_5.
- 4 , - 8 , - 16 , - 32 , \ldots
2, - 6 , 18, - 54 , \ldots
- 64 , - 16 , - 4 , -1, \ldots
State the common ratio between the terms of the following sequences:
9, 36, 144, 576, \ldots
- 6 , - 42 , - 294 , - 2058 , \ldots
2, - 16, 128, - 1024, \ldots
- 70.4 , - 17.6 , - 4.4 , - 1.1 ,\ldots
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.
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.
Find the missing terms in the following geometric progressions:
- 5, \, x, \, - 80, \, 320, \, y
a, \, b, \, \dfrac{3}{25}, \, - \dfrac{3}{125}, \, c
For each of the following pairs of terms in a geometric progression:
Find the possible values of r.
Find the value of a.
Find the general rule for T_n, for r \gt 0.
T_3 = 18 and T_5 = 162
T_4 = 32 and T_6 = 128
For each of the following recursive rules, write the rule for the nth term of the sequence:
T_{n + 1} = 6 T_n, \ T_1 = 5
T_{n + 1} = \dfrac{1}{5} T_n, \ T_1 = 213
T_n = - 4 T_{n - 1}, \ T_1 = 2
State the first 4 terms in the following recursive definition: T_n = 4 T_{n - 1}, \ T_1 = 0.5
For 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
For each of the following sequences:
Write a recursive rule for T_n in terms of T_{n - 1}, including the initial term T_1.
Find the general rule for T_n.
Calculate the 15th term of the sequence.
1, 3, 9, 27, \ldots
200, 100, 50, 25, \ldots
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} |
Consider the following sequence 5, - 10 , 20, - 40 , \ldots
Plot the first four terms on a number plane.
Is the sequence arithmetic or geometric?
Consider the sequence: 2, 6, 18, 54, \ldots
Plot the first four terms on a number plane.
Is this sequence arithmetic, geometric or neither?
Write a recursive rule for T_n in terms of T_{n - 1} and an initial condition for T_1.
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 sequence plot drawn below:
State the terms of the first five points of the sequence.
Is the sequence arithmetic or geometric? Explain your answer.
Write a recursive rule for T_{n+1} in terms of T_n and an initial condition for T_1.
For each of the following graphs, write the recursive rule for T_n in terms of T_{n - 1}, including the initial term T_1:
For each of the following:
Create a table of values for the given points.
Find r.
Write a simplified expression for T_n.
Find the 10th term of the sequence.
The plotted 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.
Suppose you save \$1 on the first day of a month, \$2 on the second day, \$4 on the third day, \$8 on the fourth day, and so on. That is, each day you save twice as much as you did the day before.
How much will you put aside for savings on the 6th day of the month?
How much will you put aside for savings on the 10th day of the month?
The average daily growth of a seedling is 10\% per day. A seedling measuring 6 \text{ cm} in height is planted.
Determine the height of the seedling at the end of Day 1.
Find the height of the seedling 2 days after it is planted.
Write a recursive rule for H_n, defining the height of the seedling n days after it is planted, and an initial condition H_0.
Radioactive element D loses half its mass every day. The initial mass of the element was 800 grams. Is the type of decay linear or exponential? Justify your answer.
Consider the table of values:
| \text{Number of days passed } (x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Population of shrimp } (y) | 5 | 25 | 125 | 625 | 3125 |
Is the number of shrimp increasing by the same amount each day?
Find the equation linking population y and time x in the form y = a^{x}.
Sketch the graph of the equation.
In a laboratory, an antibiotic is tested on a sample of 5 bacteria in a petri dish. The number of bacteria is recorded, and the bacteria are found to double each hour.
Find the equation linking the number of bacteria (y) and number of hours passed (x).