Question

If the third term of an arithmetic sequence is 11 and the eleventh term is 27, which of the following is the first term? (1 point)
Responses

16
16

2
2

-16
-16

7

Answer 1

In an arithmetic sequence, the \( n \)-th term can be expressed as:

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

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

Given that:
- The third term \( a_3 = 11 \)
- The eleventh term \( a_{11} = 27 \)

Using the formula for the third term:

\[
a_3 = a_1 + 2d = 11 \tag{1}
\]

Using the formula for the eleventh term:

\[
a_{11} = a_1 + 10d = 27 \tag{2}
\]

Now we have the following system of equations:

1. \( a_1 + 2d = 11 \)
2. \( a_1 + 10d = 27 \)

We can solve this system by subtracting equation (1) from equation (2):

\[
(a_1 + 10d) - (a_1 + 2d) = 27 - 11
\]

This simplifies to:

\[
8d = 16
\]

Now, solving for \( d \):

\[
d = 2
\]

Next, we substitute \( d \) back into equation (1) to find \( a_1 \):

\[
a_1 + 2(2) = 11
\]

This simplifies to:

\[
a_1 + 4 = 11
\]

Now, solving for \( a_1 \):

\[
a_1 = 11 - 4 = 7
\]

Thus, the first term \( a_1 \) is:

\[
\boxed{7}
\]