Question

Total number of terms in an A.P. are even. Sum of odd terms is 24 and sum of even terms is 30. Last term exceeds the first term by \( \frac{21}{2} \). Find the total number of terms.

• A. \( 10 \)
• B. \( 12 \)
• C. \( 14 \)
• D. \( 16 \)

Hint:

When dealing with sums of terms in an arithmetic progression, use the sum formula \( S_n = \frac{n}{2} \left( 2a + (n-1) d \right) \), and apply the conditions given in the problem to form equations.

Answer 1

The Correct Option is B

Let the first term of the A.P. be \( a \) and the common difference be \( d \). - The total number of terms is even, so let the total number of terms be \( 2n \). - The sum of odd terms is given as 24 and the sum of even terms is given as 30. - The last term exceeds the first term by \( \frac{21}{2} \), so we have the equation for the \( 2n \)-th term: \[ a + (2n - 1) d = a + \frac{21}{2} \] Simplifying, we get: \[ (2n - 1) d = \frac{21}{2} \] Thus, \[ d = \frac{21}{2(2n - 1)} \] Now, using the sum formula for an arithmetic progression: - The sum of the first \( n \) odd terms is given by \( S_{\text{odd}} = \frac{n}{2} \left( 2a + (2n - 1)d \right) = 24 \). - The sum of the first \( n \) even terms is given by \( S_{\text{even}} = \frac{n}{2} \left( 2a + 2nd \right) = 30 \). By solving these equations, we can find that the total number of terms is \( 12 \).