Multiple choice

The number of terms of an $A.P.$ is even; the sum of the odd terms is $24$, and of the even terms is $30$ and the last term exceeds the first by $10.5$, then the number of terms in the series is

  1. $8$
  2. $4$
  3. $6$`
  4. $10$
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A Correct answer
Explanation

Let the series contain 2m terms. The difference between the sums of even and odd-positioned terms is md, so md = 6. The last-minus-first condition gives (2m - 1)d = 10.5, yielding m = 4 and therefore 8 terms.

AI explanation

Let the number of terms be 2m, making m odd terms and m even terms. The difference between the sum of even terms and odd terms is 30 - 24 = 6, which also equals m times the common difference (d), so md = 6. The total number of terms is 2m, and the difference between the last and first term is (2m - 1)d = 10.5. Substitute d = 6/m into the second equation to get (2m - 1)(6/m) = 10.5, which simplifies to 12m - 6 = 10.5m, resulting in 2m = 8. The total number of terms is 8.

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