Multiple choice

The number of terms of an AP is even: sum of all terms at odd places and even place are 24 and 30 respectively. Last term exceeds the first term by 10.5 Then the number of terms in the series is _______.

  1. 8

  2. 12

  3. 16

  4. 20
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A Correct answer
Explanation

Let n = 2m. Sum of odd terms = m/2 * (2a + (m-1)2d) = 24. Sum of even terms = m/2 * (2(a+d) + (m-1)2d) = 30. Difference = m*d = 6. Last term - first term = (2m-1)d = 10.5. Since m*d = 6, (2m-1)d = 2md - d = 12 - d = 10.5, so d = 1.5. Then m = 6/1.5 = 4. Number of terms = 2m = 8.

AI explanation

Let the number of terms in the arithmetic progression be 2n. The difference between the sum of the even-placed terms and the odd-placed terms equals the common difference times n, giving d * n = 30 - 24 = 6. The last term exceeds the first term by 10.5, meaning (2n - 1) * d = 10.5. Substituting d = 6/n into the second equation yields (2n - 1) * (6/n) = 10.5, which simplifies to 12n - 6 = 10.5n. Solving this gives 1.5n = 6, so n = 4, making the total number of terms 2n = 8.

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