Multiple choice

Consider the sequence of numbers $2, 5, 5, 8, 8, 8, 11, 11, 11, 11, ....$ The $150^{th}$ term of the sequence is

  1. $48$
  2. $50$
  3. $47$
  4. $53$
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B Correct answer
Explanation

The sequence groups are: 2 (1 term), 5 (2 terms), 8 (3 terms), 11 (4 terms). The number of terms up to group n is n(n+1)/2. For n=16, terms = 16*17/2 = 136. For n=17, terms = 153. The 150th term is in the group of 17s. The value is 2 + (17-1)*3 = 50.

AI explanation

The number 2 appears 1 time, 5 appears 2 times, 8 appears 3 times, and 11 appears 4 times, meaning the integer k appears k times. We need the smallest integer k such that the sum of the first k natural numbers is at least 150. Using the formula for the sum of the first n natural numbers, k(k+1)/2, we test k=16 to get 16(17)/2 = 136, which covers 136 terms. Testing k=17 gives 17(18)/2 = 153, which means the 137th through 153rd terms are all 17. Because the original sequence starts at 2, the 17th distinct repeating value is 2 + (17 - 1) * 3 = 50.

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