Question

Find a general function for an arithmetic sequence with a8=14
and a14=2.



an=0+(n−1)(−1)


an=28+(n−1)(2)


an=0+(n−1)(−2)


an=28+(n−1)(−2)

Answer 1

To find the general function for the arithmetic sequence, we can use the general form of an arithmetic sequence:

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

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

We are given two terms of the sequence:

- \( a_8 = 14 \)
- \( a_{14} = 2 \)

We can express these terms using the general formula:

1. For \( a_8 \):
\[
a_8 = a_1 + (8 - 1)d = a_1 + 7d = 14 \quad \text{(Equation 1)}
\]

2. For \( a_{14} \):
\[
a_{14} = a_1 + (14 - 1)d = a_1 + 13d = 2 \quad \text{(Equation 2)}
\]

Now, we can set up a system of equations using the two equations:

\[
\begin{align*}
1. & \quad a_1 + 7d = 14 \\
2. & \quad a_1 + 13d = 2
\end{align*}
\]

Next, we can subtract Equation 1 from Equation 2 to eliminate \( a_1 \):

\[
(a_1 + 13d) - (a_1 + 7d) = 2 - 14
\]
\[
6d = -12
\]
\[
d = -2
\]

Now, substituting \( d = -2 \) back into Equation 1 to find \( a_1 \):

\[
a_1 + 7(-2) = 14
\]
\[
a_1 - 14 = 14
\]
\[
a_1 = 28
\]

Now that we have both \( a_1 \) and \( d \), we can write the general function for the sequence:

\[
a_n = a_1 + (n - 1)d
\]
\[
a_n = 28 + (n - 1)(-2)
\]

Thus, the correct answer is:

\[
\boxed{a_n = 28 + (n - 1)(-2)}
\]