The 7th term of an AP is -1 and its 16th term is 17. The nth term of the AP is

Options

A. (3n +8)
B. (4n -7)
C. (15-2n)
D. (2n-15)

Explanation

To find the nth term of the arithmetic progression (AP), we will follow these steps:

### Step 1: Understand the formula for the nth term of an AP
The nth term of an AP can be expressed using the formula:
\[ T_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Set up equations based on the given terms
We know:
- The 7th term \( T_7 = -1 \)
- The 16th term \( T_{16} = 17 \)

Using the formula for the nth term:
1. For the 7th term:
\[ T_7 = a + (7 - 1)d = a + 6d = -1 \quad \text{(Equation 1)} \]

2. For the 16th term:
\[ T_{16} = a + (16 - 1)d = a + 15d = 17 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( a + 6d = -1 \) (Equation 1)
2. \( a + 15d = 17 \) (Equation 2)

We can subtract Equation 1 from Equation 2 to eliminate \( a \):
\[ (a + 15d) - (a + 6d) = 17 - (-1) \]
This simplifies to:
\[ 15d - 6d = 17 + 1 \]
\[ 9d = 18 \]
Dividing both sides by 9 gives:
\[ d = 2 \]

### Step 4: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 6(2) = -1 \]
\[ a + 12 = -1 \]
Subtracting 12 from both sides gives:
\[ a = -1 - 12 = -13 \]

### Step 5: Write the nth term
Now that we have both \( a \) and \( d \), we can write the nth term:
\[ T_n = a + (n - 1)d = -13 + (n - 1)(2) \]
Simplifying this:
\[ T_n = -13 + 2n - 2 = 2n - 15 \]

### Final Answer
The nth term of the AP is:
\[ T_n = 2n - 15 \]

---