The third term of ann arithmetic sequence is 15, and the seventh term is 23. what is the first term?

Options

A. 1
B. 6
C. 9
D. 11

Explanation

To find the first term of the arithmetic sequence, we can follow these steps:

### Step 1: Define the terms of the arithmetic sequence
Let the first term be \( a \) and the common difference be \( d \). The \( n \)-th term of an arithmetic sequence can be expressed as:
\[ a_n = a + (n - 1)d \]

### Step 2: Write equations for the given terms
From the problem, we know:
- The third term \( a_3 \) is 15:
\[ a + 2d = 15 \quad \text{(1)} \]
- The seventh term \( a_7 \) is 23:
\[ a + 6d = 23 \quad \text{(2)} \]

### Step 3: Set up the system of equations
Now we have a system of two equations:
1. \( a + 2d = 15 \)
2. \( a + 6d = 23 \)

### Step 4: Eliminate one variable
To eliminate \( a \), we can subtract equation (1) from equation (2):
\[ (a + 6d) - (a + 2d) = 23 - 15 \]
This simplifies to:
\[ 4d = 8 \]

### Step 5: Solve for \( d \)
Now, divide both sides by 4:
\[ d = 2 \]

### Step 6: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into equation (1) to find \( a \):
\[ a + 2(2) = 15 \]
This simplifies to:
\[ a + 4 = 15 \]
Subtract 4 from both sides:
\[ a = 11 \]

### Conclusion
The first term \( a \) of the arithmetic sequence is:
\[ \boxed{11} \]