If 4 times the `4^(th)` term of an A.P. is equal to `18^(th)` times the `18^(th)` term, then find the `22^(nd)` term.

Answer

To solve the problem, we need to find the 22nd term of an arithmetic progression (A.P.) given that 4 times the 4th term is equal to 18 times the 18th term.

Let's denote:
- \( A \) as the first term of the A.P.
- \( D \) as the common difference.

### Step 1: Write the expressions for the 4th and 18th terms
The nth term of an A.P. is given by the formula:
\[ A_n = A + (n-1)D \]
Thus, the 4th term \( A_4 \) is:
\[ A_4 = A + (4-1)D = A + 3D \]
And the 18th term \( A_{18} \) is:
\[ A_{18} = A + (18-1)D = A + 17D \]

### Step 2: Set up the equation based on the problem statement
According to the problem, we have:
\[ 4 \times A_4 = 18 \times A_{18} \]
Substituting the expressions for \( A_4 \) and \( A_{18} \):
\[ 4(A + 3D) = 18(A + 17D) \]

### Step 3: Simplify the equation
Expanding both sides:
\[ 4A + 12D = 18A + 306D \]

### Step 4: Rearrange the equation
Now, let's move all terms involving \( A \) to one side and terms involving \( D \) to the other side:
\[ 4A - 18A = 306D - 12D \]
This simplifies to:
\[ -14A = 294D \]

### Step 5: Divide by -14
Dividing both sides by -14 gives:
\[ A = -21D \]

### Step 6: Find the 22nd term
Now we can find the 22nd term \( A_{22} \):
\[ A_{22} = A + (22-1)D = A + 21D \]
Substituting \( A = -21D \):
\[ A_{22} = -21D + 21D = 0 \]

### Final Answer
The 22nd term of the A.P. is:
\[ \boxed{0} \]