Multiple choice

If sum of the series $a-(a+d)+(a+2d)-(a+3d)+$......up to $21$ terms is

  1. $-10d$
  2. $a+20d$
  3. $a+10d$
  4. $20d$
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C Correct answer
Explanation

The series is a - (a+d) + (a+2d) - (a+3d) + ... up to 21 terms. This can be written as (a-a-d) + (a+2d-a-3d) + ... + (a+20d). Each pair sums to -d. There are 10 such pairs, so -10d + (a+20d) = a + 10d.

AI explanation

Group the given series into (a) + (-(a+d) + a+2d) + (-(a+3d) + a+4d) and so on, treating the first term as standalone. Since there are 21 terms, the remaining 20 terms form 10 pairs, each canceling the a and summing to a difference of +d. The total sum of these 10 pairs is 10d, and adding the initial standalone term gives a + 10d.

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