Multiple choice

If the sum of the first $40$ terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ....$ is $(102)m$ , then m is equal to :

  1. $20$
  2. $10$
  3. $5$
  4. $25$
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A Correct answer
Explanation

The series is (3+4) + (8+9) + (13+14) + ... which is 7 + 17 + 27 + ... forming an arithmetic progression with 20 terms. The sum is (n/2)(2a + (n-1)d) = (20/2)(14 + 19*10) = 10 * 204 = 2040. Given the sum is 102m, 102m = 2040, so m = 20.

AI explanation

Group the 40 terms into 10 blocks of 4 consecutive terms to analyze the pattern. The sums of these blocks form an arithmetic progression: 15, 34, 53, with a first term of 15 and a common difference of 19. The sum of these 10 blocks is calculated as (10/2) multiplied by (2(15) + 9(19)), which equals 5 multiplied by 201, giving 1005. Equating this to 102m results in m = 20.

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