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

7
7

2
2

16

Answer 1

In an arithmetic sequence, each term can be expressed in terms of the first term \( a \) and the common difference \( d \).

The \( n \)-th term of an arithmetic sequence can be given by the formula:

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

Given the information:

1. The third term \( a_3 = 11 \):
\[
a + 2d = 11 \quad \text{(1)}
\]

2. The eleventh term \( a_{11} = 27 \):
\[
a + 10d = 27 \quad \text{(2)}
\]

Now, we have a system of two equations:

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

To solve for \( a \) and \( d \), we can eliminate \( a \) by subtracting equation (1) from equation (2):

\[
(a + 10d) - (a + 2d) = 27 - 11
\]
\[
8d = 16
\]
\[
d = 2
\]

Now that we have \( d \), we can substitute it back into equation (1) to find \( a \):

\[
a + 2(2) = 11
\]
\[
a + 4 = 11
\]
\[
a = 11 - 4
\]
\[
a = 7
\]

Thus, the first term \( a \) is \( \boxed{7} \).