Consider the sequences defined by the following recursive rules. State the first 5 terms of the sequence:
a_{0} = 7 and a_{n + 1} = a_{n} + 5 for n \geq 0
a_{0} = 100 and a_{n + 1} = a_{n}-10 for n \geq 0
a_{0} = 2 and a_{n + 1} = 4 a_{n} for n \geq 0
a_{0} = 2 and a_{n + 1} = 0.5 a_{n} - 7 for n \geq 0
a_{0} = 3 and a_{n + 1} = 3 a_{n} - 3 for n \geq 0
a_{0} = 54 and a_{n + 1} = \dfrac{1}{3} a_{n} for n \geq 0
Consider the sequences defined by the following explicit rules. State the first 5 terms of the sequence:
a_{n} = 2n + 1 for n \geq 0
a_{n} = 18- 2n for n \geq 0
a_{n} = 3 n for n \geq 0
a_{n} = - 4 \left(n - 1\right) for n \geq 0
a_{n} = 2^{n + 2} for n \geq 0
a_{n} = \left( - 3 \right)^{n + 3} for n \geq 0
Consider the sequence represented in the table:
State the type of sequence.
Find x_{10}.
| n | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|
| x_{n} | - 23 | - 27 | - 31 | - 35 | - 39 |
Complete the sequences in the tables with the given conditions below:
x_{n} is arithmetic.
| n | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|
| x_{n} | 4 | 12 |
y_{n} is geometric.
| n | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|
| y_{n} | 4 | 12 |
Consider the recurrence relation u_{n + 2} = u_{n + 1} + u_{n}.
Complete the table for the first 10 terms in the sequence where u_{0} = 1 and u_{1} = 1:
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| u_{n} |
For each of the following sequences, write a recursive rule for a_{n + 1} in terms of a_{n} and an initial condition for a_{0}:
For each of the following sequences, write a formula for the n \text{th} term, a_{n}, in terms of n:
For each of the following sequences, write a recursive rule for a_{n + 1} in terms of a_{n} and an initial condition for a_{1}:
a_{n} = 7+3\left(n-1\right)
a_{n} = 8-2\left(n-1\right)
a_{n} = 2n+3
a_{n} = 8- n
a_{n} = 1.5n+6
a_{n} = - 5 + 7 n
a_{n} = 5 \left(2\right)^{n-1}
a_{n} = 20\left(1.1\right)^{n - 1}
a_{n} = 100 (0.5)^{n-1}
a_{n} = - 5 \left(4.8\right)^{n - 1}
For each of the following sequences, write a formula for the n \text{th} term of the sequence, a_{n}, in terms of n:
a_{1} = 15, a_{n + 1} = a_{n} +2 for n \geq 1
a_{1} = -9, a_{n + 1} = 2a_{n} for n \geq 1
a_{1} = - 13, a_{n + 1} = a_{n} - 8 for n \geq 1
a_{1} = 17, a_{n + 1} = - 4.7 a_{n} for n \geq 1
For each of the following, write t_n in terms of n:
t_{n + 1} = t_n - 9 where t_0 = 2.
t_{n + 1} = 2 t_{n} where t_0 = 5
t_{n + 1} = t_{n}+4 where t_{0} = 7.
t_{n + 1} = - t_{n} where t_{0} = 6.