If 7th term of an A.P. is 9 and 9th term of the A.P. is 7, then 20th term of the A.P. is

Options

A. `-2`
B. `-3`
C. `-4`
D. `-6`

Answer

To solve the problem step by step, we will use the formula for the nth term of an arithmetic progression (A.P.), which is given by:

\[ A_n = A + (n - 1)D \]

where:
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the term number.

### Step 1: Set up the equations for the 7th and 9th terms

Given:
- The 7th term \( A_7 = 9 \)
- The 9th term \( A_9 = 7 \)

Using the formula for the nth term, we can write:

1. For the 7th term:
\[ A + (7 - 1)D = 9 \implies A + 6D = 9 \quad \text{(Equation 1)} \]

2. For the 9th term:
\[ A + (9 - 1)D = 7 \implies A + 8D = 7 \quad \text{(Equation 2)} \]

### Step 2: Subtract the equations

Now, we will subtract Equation 1 from Equation 2 to eliminate \( A \):

\[ (A + 8D) - (A + 6D) = 7 - 9 \]

This simplifies to:

\[ 2D = -2 \]

### Step 3: Solve for the common difference \( D \)

Dividing both sides by 2 gives:

\[ D = -1 \]

### Step 4: Substitute \( D \) back to find \( A \)

Now that we have \( D \), we can substitute it back into Equation 1 to find \( A \):

\[ A + 6(-1) = 9 \]

This simplifies to:

\[ A - 6 = 9 \implies A = 15 \]

### Step 5: Find the 20th term

Now we can find the 20th term \( A_{20} \):

\[ A_{20} = A + (20 - 1)D = A + 19D \]

Substituting the values of \( A \) and \( D \):

\[ A_{20} = 15 + 19(-1) = 15 - 19 = -4 \]

### Final Answer

The 20th term of the A.P. is:

\[ \boxed{-4} \]