Question

If for an Arithmetic progression, 9 times the ninth term is equal to 13 times the thirteenth term, then the value of the twenty-second term is

• A. 0
• B. 2
• C. 4
• D. 5

Hint:

When given a relationship between terms in an arithmetic progression, use the general formula for the \( n \)-th term and solve for the unknowns.

Answer 1

The Correct Option is A

Step 1: Writing the general term of the Arithmetic progression.
Let the \( n \)-th term of the Arithmetic progression be given by: \[ T_n = a + (n-1)d \] where \( a \) is the first term and \( d \) is the common difference.

Step 2: Applying the condition.
We are given that 9 times the ninth term is equal to 13 times the thirteenth term: \[ 9 \cdot T_9 = 13 \cdot T_{13} \] Substituting the formula for the \( n \)-th term: \[ 9 \cdot \left( a + 8d \right) = 13 \cdot \left( a + 12d \right) \] Simplifying this equation, we find that \( a = 0 \).

Step 3: Finding the twenty-second term.
Now that we know \( a = 0 \), the twenty-second term is: \[ T_{22} = 0 + (22-1)d = 21d \] Thus, the value of the twenty-second term is 0, which makes option (A) the correct answer.