Multiple choice

The sum of certain numbers of terms of an AP series -6, -3, 0 … is 225. The number of terms is

  1. 16

  2. 15

  3. 14

  4. 13
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B Correct answer
Explanation

AP series: -6, -3, 0... a = -6, d = 3. Sum S_n = (n/2) * [2a + (n-1)d] = 225. (n/2) * [-12 + (n-1)3] = 225. n * (-12 + 3n - 3) = 450. 3n^2 - 15n - 450 = 0. n^2 - 5n - 150 = 0. (n-15)(n+10) = 0. n = 15.

AI explanation

This is an arithmetic progression with the first term a equal to negative 6 and the common difference d equal to 3. Using the sum formula for an arithmetic progression, S equals n over 2 times the quantity 2a plus n minus 1 times d, we substitute the known values to get 225 equals n over 2 times negative 12 plus 3n minus 3. Simplifying this yields 450 equals 3n squared minus 9n, and dividing by 3 gives n squared minus 3n minus 150 equals 0; factoring this quadratic equation results in n minus 15 times n plus 10 equals 0, so the positive number of terms is 15.

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