Let `S_n` denote the sum of the first n terms of an A.P. , `a_1,a_2,a_3`,…,`a_n`. If `a_5+a_9=1` and `S_9=6`, then which one of the following is not true ?

Options

A. `a_6+a_8=1`
B. `S_6=19//2`
C. `S_13=13//2`
D. `a_13=0`

Answer

To solve the problem, we need to analyze the given information about the arithmetic progression (A.P.) and derive the necessary equations.

### Given:
1. \( a_5 + a_9 = 1 \)
2. \( S_9 = 6 \)

### Step 1: Express \( a_5 \) and \( a_9 \) in terms of \( a \) and \( d \)
The general term of an A.P. can be expressed as:
\[ a_n = a + (n-1)d \]
Thus, we can write:
\[ a_5 = a + 4d \quad \text{and} \quad a_9 = a + 8d \]
Substituting these into the first equation:
\[ (a + 4d) + (a + 8d) = 1 \]
This simplifies to:
\[ 2a + 12d = 1 \quad \text{(Equation 1)} \]

### Step 2: Express \( S_9 \) in terms of \( a \) and \( d \)
The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
For \( n = 9 \):
\[ S_9 = \frac{9}{2} \times (2a + 8d) = 6 \]
Multiplying both sides by 2:
\[ 9(2a + 8d) = 12 \]
Dividing by 9:
\[ 2a + 8d = \frac{4}{3} \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( 2a + 12d = 1 \)
2. \( 2a + 8d = \frac{4}{3} \)

Subtract Equation 2 from Equation 1:
\[ (2a + 12d) - (2a + 8d) = 1 - \frac{4}{3} \]
This simplifies to:
\[ 4d = 1 - \frac{4}{3} = \frac{3}{3} - \frac{4}{3} = -\frac{1}{3} \]
Thus:
\[ d = -\frac{1}{12} \]

### Step 4: Substitute \( d \) back to find \( a \)
Substituting \( d \) into Equation 1:
\[ 2a + 12\left(-\frac{1}{12}\right) = 1 \]
This simplifies to:
\[ 2a - 1 = 1 \implies 2a = 2 \implies a = 1 \]

### Step 5: Verify the options
Now we have \( a = 1 \) and \( d = -\frac{1}{12} \).

1. **Check \( a_6 + a_8 \)**:
\[ a_6 = a + 5d = 1 + 5\left(-\frac{1}{12}\right) = 1 - \frac{5}{12} = \frac{7}{12} \]
\[ a_8 = a + 7d = 1 + 7\left(-\frac{1}{12}\right) = 1 - \frac{7}{12} = \frac{5}{12} \]
\[ a_6 + a_8 = \frac{7}{12} + \frac{5}{12} = 1 \quad \text{(True)} \]

2. **Check \( a_{13} \)**:
\[ a_{13} = a + 12d = 1 + 12\left(-\frac{1}{12}\right) = 1 - 1 = 0 \quad \text{(True)} \]

3. **Check \( S_{13} \)**:
\[ S_{13} = \frac{13}{2} \times (2a + 12d) = \frac{13}{2} \times (2 \times 1 + 12 \times -\frac{1}{12}) = \frac{13}{2} \times (2 - 1) = \frac{13}{2} \quad \text{(True)} \]

4. **Check \( a_6 \)**:
\[ a_6 = 1 + 5\left(-\frac{1}{12}\right) = \frac{7}{12} \quad \text{(Not equal to \( \frac{19}{2} \))} \]

### Conclusion
The statement that is **not true** is:
- \( a_6 \) is equal to \( \frac{19}{2} \).