Let ${n}^{th}$ term of sequence $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,.....$ is given by ${t}_{n}$ , then ?

In the given sequence, the integer k appears exactly k times, meaning the total number of terms up to the completion of the integer (k-1) is given by the sum of the first (k-1) natural numbers. To find the 100th term, we need the smallest k such that k(k+1)/2 is greater than or equal to 100. Testing k=13 gives 13(14)/2 = 91, meaning the first 91 terms are integers up to 13. Testing k=14 gives 14(15)/2 = 105, which means terms 92 through 105 are all 14. Therefore, the 100th term is 14.

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