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 the 5th and 31st terms of an arithmetic progression (AP) and the \( K^{th} \) term.

### Step-by-Step Solution:

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

2. **Setting up equations for the given terms**:
- For the 5th term (\( a_5 \)):
\[ a_5 = a + (5 - 1) \cdot d = a + 4d \]
We know \( a_5 = 1 \), so we can write:
\[ a + 4d = 1 \quad \text{(Equation 1)} \]

- For the 31st term (\( a_{31} \)):
\[ a_{31} = a + (31 - 1) \cdot d = a + 30d \]
We know \( a_{31} = -77 \), so we can write:
\[ a + 30d = -77 \quad \text{(Equation 2)} \]

3. **Subtracting the equations**:
We can eliminate \( a \) by subtracting Equation 1 from Equation 2:
\[ (a + 30d) - (a + 4d) = -77 - 1 \]
Simplifying this gives:
\[ 30d - 4d = -78 \]
\[ 26d = -78 \]
\[ d = \frac{-78}{26} = -3 \]

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

5. **Finding the \( K^{th} \) term**:
We know that the \( K^{th} \) term is given as:
\[ a_K = a + (K - 1) \cdot d \]
We are told that \( a_K = -17 \):
\[ -17 = 13 + (K - 1)(-3) \]

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

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