The `5^("th")` and the `31^("th")` terms of an arithmetic progression are, respectively 1 and `-77`. If the `K^("th")` term of the given arithmetic progression is `-17`, then the value of K is

Options

A. 12
B. 10
C. 11
D. 13

Answer

To solve the problem, we need to find the value of \( K \) given that the 5th and 31st terms of an arithmetic progression (AP) are 1 and -77, respectively, and that the \( K \)th term is -17.

### Step-by-Step Solution:

1. **Identify the general formula for the nth term of an AP**:
The nth term of an arithmetic progression can be expressed as:
\[ T_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Set up equations for the 5th and 31st terms**:
From the problem, we have:
- For the 5th term:
\[ T_5 = a + (5-1)d = a + 4d = 1 \quad \text{(1)} \]
- For the 31st term:
\[ T_{31} = a + (31-1)d = a + 30d = -77 \quad \text{(2)} \]

3. **Subtract equation (1) from equation (2)**:
To eliminate \( a \), we subtract equation (1) from equation (2):
\[ (a + 30d) - (a + 4d) = -77 - 1 \]
This simplifies to:
\[ 30d - 4d = -78 \]
\[ 26d = -78 \]
\[ d = \frac{-78}{26} = -3 \]

4. **Substitute \( d \) back into equation (1) to find \( a \)**:
Now that we have \( d \), we can substitute it back into equation (1):
\[ a + 4(-3) = 1 \]
\[ a - 12 = 1 \]
\[ a = 1 + 12 = 13 \]

5. **Write the expression for the \( K \)th term**:
Now we can express the \( K \)th term:
\[ T_K = a + (K-1)d \]
Substituting the values of \( a \) and \( d \):
\[ T_K = 13 + (K-1)(-3) \]
We know that \( T_K = -17 \):
\[ 13 + (K-1)(-3) = -17 \]

6. **Solve for \( K \)**:
Rearranging the equation:
\[ (K-1)(-3) = -17 - 13 \]
\[ (K-1)(-3) = -30 \]
Dividing both sides by -3:
\[ K-1 = 10 \]
\[ K = 10 + 1 = 11 \]

### Final Answer:
The value of \( K \) is \( 11 \).