Question

Daniel is writing an equation that can be used to find the nth term in a sequence an. In the sequence of numbers, a3=0
, a4=6
, a5=12
, a6=18
, and a7=24
.

What is the first term, a1
?
A.−6

B.−12

C.6

D.−18

Answer 1

Based on this information, which of the following is the correct equation Daniel should write?
A.an=−6n+18

B.an=−18n+6

C.an=6n−18

D.an=18n−6

Answer 2

Let's analyze the given terms in the sequence:

- \( a_3 = 0 \)
- \( a_4 = 6 \)
- \( a_5 = 12 \)
- \( a_6 = 18 \)
- \( a_7 = 24 \)

We can first see if there is a common difference between consecutive terms. The differences are:

- \( a_4 - a_3 = 6 - 0 = 6 \)
- \( a_5 - a_4 = 12 - 6 = 6 \)
- \( a_6 - a_5 = 18 - 12 = 6 \)
- \( a_7 - a_6 = 24 - 18 = 6 \)

Since the difference is consistently 6, this indicates that the sequence is an arithmetic sequence with a common difference of 6.

Now, we can express the nth term of an arithmetic sequence using the formula:

\[
a_n = a_1 + (n - 1)d
\]

where \( d \) is the common difference and \( a_1 \) is the first term.

In our case, we know:
- \( d = 6 \)

Let’s use \( a_3 \) to find \( a_1 \):

\[
a_3 = a_1 + (3 - 1) \cdot 6
\]

Since \( a_3 = 0 \), we can plug that in:

\[
0 = a_1 + 2 \cdot 6
\]
\[
0 = a_1 + 12
\]
\[
a_1 = -12
\]

Thus, the first term \( a_1 \) is \(-12\).

The correct answer is:

**B. -12**

Answer 3

To find the equation that represents the nth term of the arithmetic sequence with first term \( a_1 = -12 \) and a common difference \( d = 6 \), we use the general formula for the nth term of an arithmetic sequence:

\[
a_n = a_1 + (n - 1)d
\]

Substituting the known values:

- \( a_1 = -12 \)
- \( d = 6 \)

The equation becomes:

\[
a_n = -12 + (n - 1) \cdot 6
\]

Expanding this:

\[
a_n = -12 + 6(n - 1)
\]
\[
a_n = -12 + 6n - 6
\]
\[
a_n = 6n - 18
\]

Therefore, the correct equation Daniel should write is:

**C. \( a_n = 6n - 18 \)**.