If 5th and 6th terms of an A.P. are respectively 6 and 5, find the 11th term of the A.P.

Answer

To find the 11th term of the arithmetic progression (A.P.) where the 5th term is 6 and the 6th term is 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Terms**:
- We know that the 5th term \( T_5 = 6 \) and the 6th term \( T_6 = 5 \).

2. **Use the Formula for the n-th Term of an A.P.**:
- The formula for the n-th term of an A.P. is given by:
\[ T_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

3. **Set Up Equations for the Given Terms**:
- For the 5th term:
\[ T_5 = a + (5-1)d = a + 4d = 6 \quad \text{(Equation 1)} \]
- For the 6th term:
\[ T_6 = a + (6-1)d = a + 5d = 5 \quad \text{(Equation 2)} \]

4. **Subtract Equation 1 from Equation 2**:
- Subtracting Equation 1 from Equation 2:
\[ (a + 5d) - (a + 4d) = 5 - 6 \]
- This simplifies to:
\[ d = -1 \]

5. **Substitute the Value of d Back into Equation 1**:
- Now substitute \( d = -1 \) into Equation 1:
\[ a + 4(-1) = 6 \]
- This simplifies to:
\[ a - 4 = 6 \]
- Therefore:
\[ a = 10 \]

6. **Find the 11th Term**:
- Now, we can find the 11th term \( T_{11} \):
\[ T_{11} = a + (11-1)d = a + 10d \]
- Substitute \( a = 10 \) and \( d = -1 \):
\[ T_{11} = 10 + 10(-1) = 10 - 10 = 0 \]

### Final Answer:
The 11th term of the A.P. is \( T_{11} = 0 \).

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