If 7 times the 7th term of an AP is equal to 11 times its 11th term, then its 18th term will be

Options

A. 7
B. 11
C. 18
D. 0

Explanation

To solve the problem step-by-step, we will use the formula for the nth term of an arithmetic progression (AP). The nth term of an AP can be expressed as:

\[ a_n = a + (n - 1)d \]

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

### Step 1: Write the expressions for the 7th and 11th terms of the AP.

The 7th term (\( a_7 \)) is given by:
\[ a_7 = a + (7 - 1)d = a + 6d \]

The 11th term (\( a_{11} \)) is given by:
\[ a_{11} = a + (11 - 1)d = a + 10d \]

### Step 2: Set up the equation based on the given condition.

According to the problem, 7 times the 7th term is equal to 11 times the 11th term:
\[ 7 \cdot a_7 = 11 \cdot a_{11} \]

Substituting the expressions for \( a_7 \) and \( a_{11} \):
\[ 7(a + 6d) = 11(a + 10d) \]

### Step 3: Expand both sides of the equation.

Expanding the left side:
\[ 7a + 42d = 11(a + 10d) \]

Expanding the right side:
\[ 7a + 42d = 11a + 110d \]

### Step 4: Rearrange the equation to isolate terms involving \( a \) and \( d \).

Bringing all terms involving \( a \) to one side and terms involving \( d \) to the other side:
\[ 7a + 42d - 11a - 110d = 0 \]

This simplifies to:
\[ -4a - 68d = 0 \]

### Step 5: Solve for \( a \) in terms of \( d \).

Rearranging gives:
\[ 4a = -68d \]
\[ a = -17d \]

### Step 6: Find the expression for the 18th term.

The 18th term (\( a_{18} \)) is given by:
\[ a_{18} = a + (18 - 1)d = a + 17d \]

Substituting the value of \( a \) we found:
\[ a_{18} = -17d + 17d \]

### Step 7: Simplify to find the value of the 18th term.

This simplifies to:
\[ a_{18} = 0 \]

### Conclusion

Thus, the 18th term of the AP is:
\[ \boxed{0} \]

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