13.08 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 T_n describes the nth term of the following sequences , find the indicated term:
T_3 in the sequence: 4, - 5 , 6, - 7 , 8, \ldots
T_4 in the sequence: 200, 20, 2, 0.2, 0.02, \ldots
T_5 in the sequence: 1, 2.5, 4, 5.5, 7, 8.5, \ldots
T_3 + T_5 in the sequence: 6, - 8 , 9, - 10 , 11, \ldots
2 T_2 - T_4 in the sequence: 9, 12, 15, 18, 21, \ldots
- 4 \left(T_3 + T_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 T_n = 7.
For each of the following sequences:
Describe the 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 explicit relations:
b_n = b_{n - 1} b_{n - 2}
d_n = n^{2} + 3 n + 8
a_n = 8 a_{n - 1} + a_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
t_n = 2 n^{2} + n - 3
For the following sequencesn state the explicit rule that describes T_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
For the explicit rule T_n = 0.7 n - 5 which starts at n = 1, find the sum of the first 50 terms.
For the explicit rule T_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 T_n is the value of the nth term in the sequence, find T_{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 T_{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 T_n = 5 - 3 n, starting at n = 1:
Find T_{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.
Find the indicated term in the following cubic sequences:
8th term in the sequence: 1,8,27, \ldots
5th term in the sequence: -1,6,25, \ldots
10th term in the sequence: 2,16,54, \ldots
Consider the sequence T_n = 4n^3, starting at n = 1.
Find T_{3}.
Find the first term greater than 1000.
Consider the sequence T_n = -n^3+10, starting at n = 1.
Find T_{1}.
Find T_{4}.
Find the first term less than -300.
Consider the sequence T_n = 50 - 4n^3, starting at n = 1.
Find T_{1}.
Find T_{10}.
Find the first term less than -10\,000.