In the arithmetic sequence shown a1

Question

In the arithmetic sequence shown a1
 is the first term, a2
 is the second term, and so on. Nathaniel is writing a formula for the sequence.
 
5, 11, 17, 23, 29, 35,...

If n
 is an integer, which of these formulas represents the sequence Nathaniel could write?

Choose TWO correct answer choices.
A.an+1 = 6n+5
 for n ≥ 0

B.an = 6n+1
 for n ≥ 1

C.a1 = 5; an+1 = an+6
 for n ≥ 1

D.a1 = 5; an = an−1+6
 for n ≥ 2

E.an = 6n+1
 for n ≤ 1

F.an = 6n+1
 for n < 1

5b. 
Write the 7th
 term of the sequence.

Do not include any spaces in your answer.

GPT-4o mini · Answer

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. $a_{n + 1} = 6 n + 5$ 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. $a_{n} = 6 n + 1$ 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. $a_{n} = 6 n + 1$ 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. $a_{n} = 6 n + 1$ 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. $a_{n + 1} = 6 n + 5$ 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