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)

Question

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 \( \boxed{7} \).