Identify the following sequences as finite or infinite:
What is the third term in the sequence 4, - 5, 6, - 7, 8?
Find the next number in the sequence 16, 36, 64, 100, 144, \ldots
Find the first five terms of the following sequences. Round the terms to four decimal places where necessary.
For the sequence 3, 0.3, 0.03, 0.003, \ldots , write a formula for the nth term of the sequence, a_n, in terms of n.
Identify the following sequences as finite or infinite:
For each of the following sequences, state the first four terms:
a_1 = 4, a_n = n a_{n - 1} - 5 for n > 1.
a_1 = 3, a_n = a_{n - 1} + n for n > 1
a_1 = - 2, a_n = 2 n + 3 a_{n - 1} for n > 1.
a_1 = 3, a_2 = 5, a_n = a_{n - 1} \times a_{n - 2} for n > 2.
a_1 = 3, a_2 = 2, a_n = 3 \left(a_{n - 1}\right)^{2} - 4 a_{n - 2} for n > 2.
For each of the following sequences:
Write an initial condition for a_1.
Write a recursive rule for a_n in terms of a_{n - 1}.
-4, 0, 4, 8, \text{ . . .}
For each of the following sequences, write a formula for the nth term of the sequence, a_n, in terms of n:
a_1 = - 14, a_n = a_{n - 1} - 8 for n \geq 2
Consider the sequence a_n = 2 a_{n - 1} + 12. If a_3 = 60, find the value of a_1.
Each term of a sequence is obtained by the previous term minus 6. The first term is 3.
Write a recursive rule for T_n in terms of T_{n - 1} and an initial condition for T_1 that defines this sequence.
For each of the following rules:
Write the first term of the sequence defined by this rule, if the starting number is 8.
Write an equation for the term, T, in terms of the starting number, n.
The starting number is doubled, then 4 is subtracted.
The starting number has 9 added to it. The sum is then multiplied by 5.
Matches were used to make the pattern attached:
Complete the table:
| \text{Number of triangles }(t) | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| \text{Number of matches }(m) |
Write a formula that describes the relationship between the number of matches, m, and the number of triangles, t.
How many matches are required to make 73 triangles using this pattern?
Matches were used to make the pattern attached:
Complete the table:
| \text{Number of triangles }(t) | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| \text{Number of matches }(m) |
Write a formula that describes the relationship between the number of matches, m, and the number of triangles, t.
How many matches are required to make 53 triangles using this pattern?
Consider the pattern for blue boxes shown below:
Complete the table:
| \text{Number of columns }(c) | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| \text{Number of blue boxes }(b) |
Write a formula that describes the relationship between the number of blue boxes, b and the number of columns, c.
How many blue boxes will there be if this pattern were to continue for 92 columns?
If this pattern continued and we had 51 blue boxes, how many columns would we have?
Consider the pattern below:
If s is the total number of squares, write an expression that gives s in terms of n.
Find the number of squares when n = 9.
Consider the pattern of images below:
If p represents the number of shaded triangles, write an expression for p in terms of n.
Find the value of p when n = 8.
Consider the pattern below:
Find the number of circles, m, when n = 7.
For each of the following sequences:
Is the sequence a Fibonacci-type sequence?
What are the next two terms of the sequence?
Use the fact that the Fibonacci sequence is defined by t_n = t_{n - 2} + t_{n - 1}, where t_1 = 1 and t_2 = 1, to find the first 10 terms.
State the term in the Fibonacci sequence represented by:
Find the first ten terms of a sequence defined by t_n = t_{n - 2} + t_{n - 1}, where t_1 = 1 and t_2 = 3.
Explain how the Fibonacci sequence can be found using Pascal's triangle.
The Fibonacci sequence starts with 1,1,2..., so that t_1 = 1. Which is the first term in the Fibonacci sequence that is greater than 300?
In the Fibonacci sequence, t_{21} = 10\,946, t_{23} = 28\,657 and t_{24} = 46\,368. Find:
t_{25}
t_{19} + t_{20}
t_{22}
In the Fibonacci sequence, t_{18} = 2584, t_{19} = 4181 and t_{21} = 10\,946. Find:
t_{20}
t_{17}
t_n is a term in the Fibonacci sequence. State the term in the Fibonacci sequence represented by t_{70} + 2 t_{71} + t_{72}.
Use the fact that the nth term of the Fibonacci sequence is given by:
\dfrac{1}{\sqrt{5}} \left(\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n} - \left(\dfrac{1 - \sqrt{5}}{2}\right)^{n}\right)
to find the value of the 18th term of the sequence.
Each day, Kumi withdraws \$4 from her bank account to spend on lunch. Before she withdraws this amount on January 1, she has \$2000 in her bank account.
After Kumi withdraws \$4 on January 7, how much does she have left in her bank account?
How much does Kumi have left in her bank account at the end of February 4?
Write a recursive rule for t_n in terms of t_{n - 1} which defines the amount Kumi has in her account at the end of day n, and an initial condition for t_0.
Zuber is a taxi service that charges a \$1.50 pick-up fee and \$1.95 per kilometre of travel.
What is the total charge for a 10 km journey?
Write a recursive rule for T_n in terms of T_{n - 1} which defines the price of a n km trip, and an initial condition for T_0.
Christa deposits \$1 into a new bank account while Jenny deposits \$38 into a new bank account. The next day, Christa adds \$2 to the account, and doubles how much she adds each day thereafter. Jenny adds \$5 each day.
Fill in the amount Christa had in the account each day for the first four days.
Fill in the amount Jenny had in the account each day for the first four days.
On which day will they have the same amount of money?
For a photo, the staff at a company have been arranged such that there are 9 people in the front row and each row has 2 more people than in the row in front of it.
Write a recursive rule for a_n, the number of people in the nth row, in terms of a_{n - 1} and an initial condition for a_1.
Write a formula for a_n in terms of n.
How many people are in the 10th row?
A basketball is dropped onto the ground from a height of 15 metres. On each bounce, the ball reaches a maximum height of 55\% of its previous maximum height.
Write a recursive rule for a_n, the height of the ball on the nth bounce, in terms of a_{n - 1} and an initial condition a_0.
Write a formula for a_n, for the height reached on the nth bounce in terms of n.
How high does the basketball reach after the 5th bounce? Round your answer to two decimal places.