Question

An arithmetic sequence has these terms: a Subscript 2 Baseline equals 13, a Subscript 4 Baseline equals 25, a Subscript 6 Baseline equals 37, and a Subscript 8 Baseline equals 49.



Which formula can be used to find a Subscript n Baseline?

Answer options with 5 options
A.
a Subscript n Baseline equals 13 plus 12 left-parenthesis n minus 1 right-parenthesis

B.
a Subscript n Baseline equals 13 plus 6 left-parenthesis n minus 1 right-parenthesis

C.
a Subscript n Baseline equals 7 plus 12 left-parenthesis n minus 1 right-parenthesis

D.
a Subscript n Baseline equals 7 plus 6 left-parenthesis n minus 1 right-parenthesis

E.
a Subscript n Baseline equals 6 plus 7 left-parenthesis n minus 1 right-parenthesis

Answer 1

To find the formula for the \(n\)-th term \(a_n\) of the given arithmetic sequence, we first need to identify the common difference and the first term of the sequence.

We have the terms:
- \(a_2 = 13\)
- \(a_4 = 25\)
- \(a_6 = 37\)
- \(a_8 = 49\)

Let's calculate the common difference \(d\) using the first two terms we have:

1. \(a_4 - a_2 = 25 - 13 = 12\)
- Since \(a_4\) is the 4th term and \(a_2\) is the 2nd term, two terms apart indicates that the difference will be \(d \cdot 2 = 12\). Hence, \(d = 12/2 = 6\).

Now that we have the common difference \(d = 6\), we can determine a formula for the \(n\)-th term. The general formula for an arithmetic sequence is:

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

We already know \(d = 6\). We need to find \(a_1\), the first term of the sequence.

We can find \(a_1\) by using \(a_2\):
\[
a_2 = a_1 + (2-1)d \implies 13 = a_1 + 1 \cdot 6 \implies a_1 + 6 = 13 \implies a_1 = 7
\]

Now we can substitute \(a_1\) and \(d\) into the formula:

\[
a_n = 7 + (n-1)6
\]

Now simplifying this gives:

\[
a_n = 7 + 6(n-1) \implies a_n = 7 + 6n - 6 \implies a_n = 6n + 1
\]

So the formula to find \(a_n\) is:

\[
a_n = 7 + 6(n - 1)
\]

Looking at the provided options, we see that option D matches our derived formula.

**The correct answer is:**

**D. \(a_n = 7 + 6(n - 1)\)**