Multiple choice

The $100^{th}$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...,$ is

  1. $12$
  2. $13$
  3. $14$
  4. $15$
  5. $16$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sequence is 1, 2, 2, 3, 3, 3, 4, 4, 4, 4... The number n appears n times. The total count of terms up to number k is k(k+1)/2. For k=13, the count is 91. The 100th term will be 14.

AI explanation

We must find the smallest integer k such that the sum of the first k integers is at least 100, which gives the inequality k(k+1)/2 >= 100. Solving k^2 + k - 200 >= 0 yields k = 14, since 14(15)/2 = 105 and 13(14)/2 = 91. The first 13 numbers occupy 91 positions, meaning positions 92 through 105 are all occupied by the number 14. Therefore, the 100th term is 14.

Keep practicing — related questions