Multiple choice

How many terms of the series 54,51,48, ... be taken so that their sum is 513?

  1. only 18

  2. only 19

  3. only 17

  4. both 18 and 19
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Sum of AP = n/2 * (2a + (n-1)d). 513 = n/2 * (108 + (n-1)(-3)). 1026 = n * (108 - 3n + 3) = n * (111 - 3n). 1026 = 111n - 3n^2. 3n^2 - 111n + 1026 = 0. n^2 - 37n + 342 = 0. (n-18)(n-19) = 0. Both 18 and 19 are solutions.

AI explanation

Using the arithmetic progression sum formula S = n/2 * (2a + (n - 1)d), we substitute the first term a = 54, common difference d = -3, and sum S = 513. This yields 513 = n/2 * (108 + (n - 1)(-3)), which simplifies to 513 = n/2 * (111 - 3n). Multiplying by 2 and rearranging gives the quadratic equation 3n^2 - 111n + 1026 = 0. Dividing by 3 results in n^2 - 37n + 342 = 0, which factors into (n - 18)(n - 19) = 0. Therefore, the number of terms can be either 18 or 19, meaning both values are correct.

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