The seventh term of an arithmetic sequence is 5 and the twelfth term -15. The first term of this sequence is

Options

A. 20
B. 29
C. 30
D. 31

Explanation

To find the first term of the arithmetic sequence given that the seventh term is 5 and the twelfth term is -15, we can follow these steps:

### Step 1: Write the formula for the nth term of an arithmetic sequence
The nth term of an arithmetic sequence can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( T_n \) is the nth term, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

### Step 2: Set up equations for the given terms
For the 7th term:
\[ T_7 = a + (7 - 1) \cdot d = a + 6d \]
Given that \( T_7 = 5 \), we can write:
\[ a + 6d = 5 \quad \text{(1)} \]

For the 12th term:
\[ T_{12} = a + (12 - 1) \cdot d = a + 11d \]
Given that \( T_{12} = -15 \), we can write:
\[ a + 11d = -15 \quad \text{(2)} \]

### Step 3: Subtract the two equations
Now, we will subtract equation (1) from equation (2):
\[ (a + 11d) - (a + 6d) = -15 - 5 \]
This simplifies to:
\[ 11d - 6d = -20 \]
\[ 5d = -20 \]

### Step 4: Solve for the common difference \( d \)
Now, divide both sides by 5:
\[ d = \frac{-20}{5} = -4 \]

### Step 5: Substitute \( d \) back into one of the original equations
Now that we have \( d \), we can substitute it back into equation (1) to solve for \( a \):
\[ a + 6(-4) = 5 \]
This simplifies to:
\[ a - 24 = 5 \]
Adding 24 to both sides gives:
\[ a = 5 + 24 = 29 \]

### Final Answer
The first term of the arithmetic sequence is:
\[ \boxed{29} \]