Describe how the common ratio of a geometric sequence is obtained.
If T_n is the nth term for the following geometric sequences, evaluate:
\dfrac{T_2}{T_1}
\dfrac{T_3}{T_2}
\dfrac{T_4}{T_3}
T_5
- 4 , - 8 , - 16 , - 32 , \ldots
2, - 6 , 18, - 54 , \ldots
- 64 , - 16 , - 4 , -1, \ldots
Determine the next term of the sequence -1, - 7, - 49, \ldots
Determine the next three terms of the sequence - 1, 2, -4, \ldots
Calculate the common ratio for each of the following geometric sequences:
9, 36, 144, 576, \ldots
- 6 , - 42 , - 294 , - 2058 , \ldots
Find the missing terms in the following geometric sequence:
- 5, x, - 80, 320, y
Find the first three terms in the following geometric progressions:
The first term is 6 and the common ratio is 4.
The first term is 7 and the common ratio is - 2.
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 sequences:
Write a recursive rule for T_n in terms of T_{n - 1}, including the initial term T_1.
Write the explicit rule for T_n, the nth term of this sequence.
Calculate the 15th term of the sequence. Write your answer in index form.
1, 3, 9, 27, \ldots
200, 100, 50, 25, \ldots
For each of the following recursive rules, write the explicit rule for the nth term of the sequence:
T_{n + 1} = 6 T_n,\text{ } T_1 = 5
T_{n + 1} = \dfrac{1}{5} T_n,\text{ } T_1 = 213
T_n = - 4 T_{n - 1},\text{ } T_1 = 2
Consider the sequence 2, 6, 18, 54, \ldots
Plot the first four terms on a graph.
Is this sequence arithmetic or geometric?
Write a recursive rule for T_{n+1} in terms of T_n and an initial condition for T_1.
Each of the following graphs contain the first 4 terms of a sequence.
For each sequence, write a recursive rule for T_n in terms of T_{n-1} including the initial term T_1.
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.
In a geometric progression, T_3 = 18 and T_5 = 162.
Find the possible values of r, the common ratio in the sequence.
Find the value of a, the first term in the progression.
If r is positive, determine the general rule for T_n.
In a geometric progression, T_4 = 32 and T_6 = 128.
Find the possible values of r.
If r = 2, find a.
Hence, determine the general rule for T_n.
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. So, each day you save twice as much as you did the day before.
Calculate how much you will save:
On the 6th day of the month?
On the 10th day of the month?
The average daily growth of a seedling is 10\% per day. A seedling measuring 6 cm in height is planted.
What is the height of the seedling at the end of day 1?
What is the height of the seedling 2 days after it is planted?
Write a recursive rule for H_n, for the height of the seedling n days after it is planted, and an initial condition H_0.
The following table shows the mass of 800 grams of radioactive element D left each day, given that element D loses half its mass every day.
Complete the table.
What type of decay is this, linear or exponential?
| Day | Mass of element D (g) |
|---|---|
| 0 | 800 |
| 1 | |
| 2 | |
| 3 | |
| 4 |
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}.
Graph the equation on a number plane.
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.
Complete the table below:
| \text{Number of hours passed }(x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Number of bacteria }(y) | 5 | 20 |
If y represents the number of bacteria at time x, which general equation satisfies this model?
y = - a^{ - x }
y = b a^{ - x }
y = b \times a^{x}
y = a^{x}
Find the equation linking the number of bacteria, y, and the number of hours passed, x.
How many bacteria will be present in the petri dish after 18 hours? Write your answer in index form.