To analyze the given arithmetic sequence: 5, 11, 17, 23, 29, 35,... we can identify that the first term \( a_1 \) is 5 and the common difference \( d \) is 6.
Now, let's examine the formula options given:
A. \( a_{n+1} = 6n + 5 \) for \( n \geq 0 \)
If we substitute \( n = 0 \):
\( a_1 = 6(0) + 5 = 5 \) (This is correct)
If we substitute \( n = 1 \):
\( a_2 = 6(1) + 5 = 11 \) (This is correct)
If we substitute \( n = 2 \):
\( a_3 = 6(2) + 5 = 17 \) (This is correct)
Thus, this option is correct.
B. \( a_n = 6n + 1 \) for \( n \geq 1 \)
If we substitute \( n = 1 \):
\( a_1 = 6(1) + 1 = 7 \) (This is incorrect)
This option is not correct.
C. \( a_1 = 5; a_{n+1} = a_n + 6 \) for \( n \geq 1 \)
Starting from \( a_1 = 5 \):
\( a_2 = 5 + 6 = 11 \) (This is correct)
\( a_3 = 11 + 6 = 17 \) (This is correct)
Thus, this option is correct.
D. \( a_1 = 5; a_n = a_{n-1} + 6 \) for \( n \geq 2 \)
Starting from \( a_1 = 5 \):
\( a_2 = a_1 + 6 = 5 + 6 = 11 \) (This is correct)
\( a_3 = a_2 + 6 = 11 + 6 = 17 \) (This is correct)
Thus, this option is correct.
E. \( a_n = 6n + 1 \) for \( n \leq 1 \)
If we substitute \( n = 1 \):
\( a_1 = 6(1) + 1 = 7 \) (This is incorrect)
Thus, this option is not correct.
F. \( a_n = 6n + 1 \) for \( n < 1 \)
This formula cannot provide any terms of the sequence correctly and is inappropriate as the definition is invalid for \( n \geq 1\).
Thus, this option is also not correct.
From the analysis, the two correct answer choices are:
- A. \( a_{n+1} = 6n + 5 \) for \( n \geq 0 \)
- C. \( a_1 = 5; a_{n+1} = a_n + 6 \) for \( n \geq 1 \)
Now, for the 7th term of the sequence:
To find the 7th term \( a_7 \): Using the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n - 1)d \] Where:
- \( a_1 = 5 \)
- \( d = 6 \)
- \( n = 7 \)
Calculating \( a_7 \): \[ a_7 = 5 + (7 - 1) \times 6 \] \[ a_7 = 5 + 6 \times 6 \] \[ a_7 = 5 + 36 \] \[ a_7 = 41 \]
Thus, the answer for the 7th term is: 41