8.01 Sequences
For each of the following sequences, find the indicated term:
Third term in the sequence: 2, - 4 , 6, - 8 , 16, \ldots
Fourth term in the sequence: 3, 3.5, 4, 4.5, 5, 5.5, \ldots
Fifth term in the sequence: 5, 4, 3, 2, 1, 0, - 1 , \ldots
If u_n describes the nth term of the following sequences , find the indicated term:
u_3 in the sequence: 4, - 5 , 6, - 7 , 8, \ldots
u_4 in the sequence: 200, 20, 2, 0.2, 0.02, \ldots
u_5 in the sequence: 1, 2.5, 4, 5.5, 7, 8.5, \ldots
u_3 + u_5 in the sequence: 6, - 8 , 9, - 10 , 11, \ldots
2 u_2 - u_4 in the sequence: 9, 12, 15, 18, 21, \ldots
- 4 \left(u_3 + u_4\right) in the sequence: 1, 4, 5, 9, 14, 23, \ldots
In the sequence 5, - 5 , 7, - 7 , 9, - 9 , \ldots , find n if u_n = 7.
For each of the following sequences:
Describe the recurring pattern in words.
Find the next number in the sequence.
- 1 , 1, 3, 5, 7, \ldots
64, 32, 16, 8, 4, 2, \ldots
2, - 4 , 6, - 8 , 10, - 12 , \ldots
Determine whether the following are recurrence relations:
\dfrac{S_n}{n} = n + 1
R_n = \left( 3 R_{n - 1}\right)^{5} + R_{n - 2}
u_n = u_{n - 1} + 9
V_n = 3 \left(n - 1\right)
Consider the following sequence given in the table:
Write down the general rule for u_n in terms of n.
Use the general rule to find u_{20}.
| n | 1 | 2 | 3 | 4 | \ldots |
|---|---|---|---|---|---|
| u_n | 1 | 4 | 9 | 16 | \ldots |
Consider the following sequence given in the table:
Write down the general rule for u_n in terms of n.
Use the general rule to find u_{19}.
| n | 1 | 2 | 3 | 4 | \ldots |
|---|---|---|---|---|---|
| u_n | 1 | 8 | 27 | 27 | \ldots |
Write the recursive rule for each of the given description. Let u_n be the nth term.
To find the next term, add 5 to the previous term.
To find the next term, add the two previous terms.
To find the next term, multiply the previous term by negative 1 and then add 7.
To find the next term, subtract one and a half from the previous term.
To find the next term, subtract 6 from the previous term.
To find the next term, multiply the two previous terms.
To find the next term, multiply the previous term by negative 5 and then subtract 3.
To find the next term, add three and a quarter to the previous term.
For each of the following recursive rules, state the first 5 terms of the sequences in order:
u_n = 2 u_{n - 1},\ u_1 = 2
u_n = u_{n - 1} + 6,\ u_1 = - 8
b_n = - b_{n - 1} + 3,\ b_1 = 0
For each of the recursive relations below, find the following.
u_2
u_3
u_4
u_n = \left( - 1 \right)^{n + 1} u_{n - 1},\ u_1 = 2
u_{n + 1} = \left( 2 u_n\right)^{n - 1},\ u_1 = 3
u_n = n^{n + 1} + u_{n - 1},\ u_1 = 3
u_{n + 1} = \left( 4 u_n\right)^{n - 1} \times \left(n + 1\right),\ u_1 = 5
Consider the sequence which has a first term of 11 and a second term of 22, and subsequent terms are found by adding the two previous terms.
Write the recursive rule for the sequence, letting u_n be the n\text{th} term.
State the first five terms of the sequence.
Determine whether the following are explicit relations:
b_n = b_{n - 1} b_{n - 2}
d_n = n^{2} + 3 n + 8
u_n = 8 u_{n - 1} + u_1
c_n = 3 n^{2}
For each of the following explicit rules, state the first 5 terms of the sequences in order starting with n=1:
b_n = 5 n - 2
s_n = n^{2} + 6
u_n = 2 n^{2} + n - 3
For the following sequences:
State the recursive rule that describes u_n in terms of u_{n - 1}, and the initial term u_1.
State the explicit rule that describes u_n in terms of n.
A sequence starts with a first term of 1300 and each subsequent term increases by 2.5\% of the previous term.
A sequence starts with a first term of 44 and each term is 77 more than the previous term.
Find the indicated term in the following sequences:
21st term in the sequence: - 3 , - 6 , - 9 , - 12 , - 15 , \ldots
53rd term in the sequence: 2, 3.5, 5, 6.5, 8, 9.5, \ldots
39th term in the sequence: 10, 8, 6, 4, 2, 0, - 2 , \ldots
Consider the sequence defined by u_n = 2 u_{n - 1} + 3, u_1 = 12. Calculate the sum of the first 25 terms.
For each of the following pairs of sequences, compare the first 10 terms of each sequence on a graph to determine which sequence is increasing faster:
u_{n + 1} = u_n + 22,\ u_1 = 12
A_{n + 1} = A_n + 25,\ A_1 = 12
u_{n + 1} = 2.5\, u_n,\ u_1 = 5
A_{n + 1} = 3.5 A_n,\ A_1 = 5
For each pair of sequences, compare the first 10 terms of each sequence on a graph to determine which sequence is decreasing faster:
u_{n + 1} = u_n - 5,u_1 = 4 and A_{n + 1} = A_n - 10, A_1 = 12
u_{n + 1} = 0.2 u_n, u_1 = 20 and A_{n + 1} = 0.6 A_n, A_1 = 20
For the explicit rule u_n = 0.7 n - 5 which starts at n = 1, find the sum of the first 50 terms.
For the explicit rule u_n = \left( 0.2 n\right)^{2} + 5 n which starts at n = 1, find the sum of the terms from the \\20th to the 30th term inclusive.
Consider the sequence 3, 7, 11, 15, 19, 23, \ldots
If u_n is the value of the nth term in the sequence, find u_{10}.
Find the first term greater than 100.
If S_n is the sum of the first n terms, find S_{10}.
Starting from n = 1, find the minimum number of terms for the sum to first exceed 500.
Consider the sequence 1, 2, 4, 8, 16, \ldots
Find u_{13}.
Find the first term greater than 10\,000.
If S_n is the sum of the first n terms, find S_{20}.
Starting from n = 1, find the greatest number of terms such that the sum is still less than 10\,000\,000.
Consider the sequence u_n = u_{n - 1} + 5, u_1 = 10
Find u_{40}.
Find the first term greater than or equal to 300.
If S_n is the sum of the first n terms, find S_{30}.
Starting from n = 1, find the minimum number of terms for the sum to be greater than 5000.
Consider the sequence u_{n+1} = 1.5 u_n, u_1 = 10
Find u_{10} to the nearest whole number.
Find the first term greater than or equal to 1000. Round your answer to the nearest whole number.
If S_n is the sum of the first n terms, find S_{18}. Round your answer to the nearest whole number.
Starting from n = 1, find the minimum number of terms for the sum to be greater than 50\,000.
Consider the sequence u_n = 5 - 3 n, starting at n = 1:
Find u_{30}.
Find the first term less than - 150.
If S_n is the sum of the first n terms, find S_{15}.
Starting from n = 1, find the minimum number of terms required for the sum to be less than - 600.
Consider the sequence u_n = 200 \times 0.5^{n - 1}
Find u_8.
Find the first term less than 0.5.
If S_n is the sum of the first n terms, find S_{10}.
Starting from n = 1, find the minimum number of terms for the sum to first exceed 399.9.