Multiple choice

The sum of the series a - (a + d) + (a + 2d) - (a + 3d) + . . . upto (2n + 1) terms is

    • nd

  2. a + 2nd

  3. a + nd

  4. 2nd
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C Correct answer
Explanation

Series: a - (a+d) + (a+2d) - (a+3d) + ... + (a+2nd). Grouping in pairs: (a - a - d) + (a + 2d - a - 3d) + ... = -d - d - ... (n times) + (a + 2nd). Total = a + 2nd - nd = a + nd.

AI explanation

The given series is an arithmetic progression with alternating signs, which means the even terms are subtracted and the odd terms are added. Grouping the series into pairs of one negative and one positive term leaves the last term unpaired because there are (2n + 1) terms. The sum of the first 2n paired terms is -nd, so adding the final odd term, which is the (2n + 1)th term equal to a + 2nd, gives a + nd.

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