Multiple choice

If the sum of the first n terms of the series $\sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+....$ is $435\sqrt{3}$, then n equals.

  1. $18$
  2. $13$
  3. $29$
  4. $15$
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D Correct answer
Explanation

The series is sqrt(3) * (1 + 5 + 9 + 13 + ...). This is an AP with a=1 and d=4. Sum = (n/2) * (2*1 + (n-1)*4) = (n/2) * (4n - 2) = n(2n - 1). Setting n(2n - 1) = 435, we get 2n^2 - n - 435 = 0. Solving gives n = 15.

AI explanation

First, simplify the terms of the series by factoring out the square roots to get 3 to the power of one-half, 5 times 3 to the power of one-half, 9 times 3 to the power of one-half, and so on. Dividing the total sum by 3 to the power of one-half gives the arithmetic progression 1 + 5 + 9 + 13 + ..., which has a first term of 1 and a common difference of 4. Setting the sum formula for n terms equal to 435 yields n squared equals 225, so n is 15.

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