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

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:

\[ \boxed{7} \]