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.

Answer 1

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