Multiple choice

Following two given series are in A.P. $2, 4, 6, 8 ...$ $3, 6, 9, 12 ...$ First series contains $30$ terms, while the second series contains $20$ terms. Both of the above given series contains some terms, which are common to both of them. The sum of both the above given A.P. are respectively:

  1. $(930, 630)$
  2. $(630, 930)$
  3. $(870, 580)$
  4. $(580, 870)$
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A Correct answer
Explanation

Series 1: 2, 4, 6... (30 terms). Sum = (30/2)(2*2 + 29*2) = 15(4+58) = 15*62 = 930. Series 2: 3, 6, 9... (20 terms). Sum = (20/2)(2*3 + 19*3) = 10(6+57) = 10*63 = 630.

AI explanation

Using the formula for the sum of an arithmetic progression, S = n/2 times (2a + (n-1)d), we calculate the sums for the two series. For the first series of 30 terms, the sum is 30/2 times (2(2) + 29(2)), which equals 15 times 62, giving 930. For the second series of 20 terms, the sum is 20/2 times (2(3) + 19(3)), which equals 10 times 63, giving 630.

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