The first, second and last terms of an AP are `4, 7 and 31` respectively, then

Options

A. the third term is 15
B. the number of terms is 10
C. the sum of the terms is 155
D. None of these

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the terms of the AP
Given:
- First term \( a = 4 \)
- Second term \( a_2 = 7 \)
- Last term \( a_n = 31 \)

### Step 2: Calculate the common difference \( d \)
The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 7 - 4 = 3 \]

### Step 3: Find the number of terms \( n \)
We know the formula for the last term of an AP:
\[ a_n = a + (n-1)d \]
Substituting the known values:
\[ 31 = 4 + (n-1) \cdot 3 \]
Now, simplify and solve for \( n \):
\[ 31 - 4 = (n-1) \cdot 3 \]
\[ 27 = (n-1) \cdot 3 \]
\[ n - 1 = \frac{27}{3} = 9 \]
\[ n = 9 + 1 = 10 \]

### Step 4: Calculate the third term \( a_3 \)
The third term can be calculated using the formula:
\[ a_3 = a + (3-1)d = 4 + 2 \cdot 3 = 4 + 6 = 10 \]

### Step 5: Calculate the sum of the first \( n \) terms \( S_n \)
The sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]
Substituting the known values:
\[ S_{10} = \frac{10}{2} \cdot (2 \cdot 4 + (10-1) \cdot 3) \]
Calculating:
\[ S_{10} = 5 \cdot (8 + 9 \cdot 3) = 5 \cdot (8 + 27) = 5 \cdot 35 = 175 \]

### Summary of Results
- Third term \( a_3 = 10 \)
- Number of terms \( n = 10 \)
- Sum of terms \( S_n = 175 \)

### Conclusion
The correct answer is:
- The third term is **10** (not correct as per the question)
- The number of terms is **10** (correct)
- The sum of terms is **175** (not correct as per the question)