Given the following sequences, find the indicated term:
The 53^\text{rd} term in: 2, 3.5, 5, 6.5, 8, 9.5, \ldots
The 21^\text{st} term in: - 3 , - 6 , - 9 , - 12 , - 15 , \ldots
The 39^\text{th} term in: 10, 8, 6, 4, 2, 0, - 2 , \ldots
Calculate:
T_3 + T_5 for the sequence: 6, - 8 , 9, - 10 , 11, \ldots
2 T_2 - T_4 for the sequence: 9, 12, 15, 18, 21, \ldots
- 4 \left(T_3 + T_4\right) for the sequence: 1, 4, 5, 9, 14, 23, \ldots
Consider the sequence 5, - 5 , 7, - 7 , 9, - 9 , \ldots Find n if T_n = 7.
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 T_n be the n\text{th} term.
State the first five terms of the sequence.
For the following recursive relations, find:
T_2
T_3
T_4
T_n = n^{n + 1} + T_{n - 1}, T_1 = 3
T_{n + 1} = \left( 4 T_n\right)^{n - 1} \times \left(n + 1\right), T_1 = 5
Consider the sequence defined by a_n = 2 a_{n - 1} + 3, a_1 = 12. Calculate the sum of the first 25 terms.
Consider the sequence defined by b_{n+1} = 1.8 b_n + 50, b_1 = 1. Calculate the sum of the terms from the 20th term to the 29th term inclusive. Round your answer to the nearest whole number.
Consider the sequence T_n = T_{n - 1} + 5, T_1 = 10
Find T_{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 T_{n+1} = 1.5 T_n, T_1 = 10
Find T_{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.
For each pair of sequences, compare the first 10 terms of each of these sequence on a graph to determine which sequence is increasing faster:
T_{n + 1} = 2.5 T_n, T_1 = 5 and A_{n + 1} = 3.5 A_n, A_1 = 5
T_{n + 1} = T_n + 22,T_1 = 12 and A_{n + 1} = A_n + 25, A_1 = 12
For each pair of sequences, compare the first 10 terms of each of these sequence on a graph to determine which sequence is decreasing faster.
T_{n + 1} = T_n - 5,T_1 = 4 and A_{n + 1} = A_n - 10, A_1 = 12
T_{n + 1} = 0.2 T_n, T_1 = 20 and A_{n + 1} = 0.6 A_n, A_1 = 20
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 given in the table below:
Use the general rule to find T_{20}
| n | 1 | 2 | 3 | 4 | ... |
|---|---|---|---|---|---|
| T_n | 1 | 4 | 9 | 16 | ... |
Consider the sequence given in the table below:
Use the general rule to find T_{19}.
| n | 1 | 2 | 3 | 4 | ... |
|---|---|---|---|---|---|
| T_n | 1 | 8 | 27 | 64 | ... |
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.
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 = 200 \times 0.5^{n - 1}
Find T_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.
For the following sequences:
State the recursive rule that describes T_n in terms of T_{n - 1}, and the initial term T_1.
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.