Multiple choice

If the sum of the first $15$ terms of the series $3+7+14+24+37+.......$ is $15k$, then $k$ is equal to:

  1. $122$
  2. $81$
  3. $119$
  4. $126$
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A Correct answer
Explanation

The series is 3, 7, 14, 24, 37. The differences are 4, 7, 10, 13 (an AP). The n-th term is a quadratic: T_n = (3/2)n^2 - (1/2)n + 2. Summing this for 15 terms gives 1830. 1830 / 15 = 122.

AI explanation

The differences between consecutive terms are 4, 7, 10, and 13, which form an arithmetic progression with a first term of 4 and a common difference of 3. The sum of the first 15 terms of this series can be calculated using the standard sum formula for a series with linearly increasing differences, yielding a total sum of 1830. Since this total is given as 15k, we divide 1830 by 15 to find the value of the variable. The result is 122.

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