Multiple choice

How many terms of the series $54+51+48+45+.......$ must be taken to make $513$?

  1. only $n=18$
  2. only $n=19$
  3. only $n=17$
  4. both $n=18,19$
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D Correct answer
Explanation

Sum = n/2 * [2a + (n-1)d]. 513 = n/2 * [108 + (n-1)(-3)] = n/2 * [108 - 3n + 3] = n/2 * [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 work because the 19th term is 54 + 18*(-3) = 0.

AI explanation

Using the arithmetic progression sum formula n divided by 2 multiplied by the quantity 2a plus the quantity n minus 1 times d, we set the sum equal to 513 with the first term as 54 and the common difference as -3. This gives 513 equals n divided by 2 multiplied by the quantity 108 minus 3 times the quantity n minus 1, which simplifies to 1026 equals 111n minus 3n squared. Factoring the quadratic equation 3n squared minus 111n plus 1026 equals 0 gives two possible values for n, which are 18 and 19, so the result is both n=18,19.

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