<s>[INST] What is the 94th term of this sequence? $1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,\ldots$ 
What is the 94th term of the following sequence?
  $$1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,\ldots$$
  
  
*
  
*8
  
*9
  
*10
  
*11

My Attempt: I found that the answer is 3rd option i.e. 94th term is 10. As every number is written 2n: n is natural number. Here  94 = 2(47)  so sum of first few natural numbers should be greater than or equal to 47. Since $$1+2+3+4+5+6+7+8+9 = 45 < 47$$  so options 1,2 are not possible and $$1+2+3+4+5+6+7+8+9+10 = 55 >47$$ But this is a lengthy process.
Please tell me easiest way to approach the answer.
 [/INST] To detremine the $94$-th term, I can use the rules of the aritmethic progression, in particular: $$S_n=\frac{n}{2}(2a_0+(n-1)d)$$ Substituing the numbers, I have: $$S_n=\frac{n}{2}(2+2n)$$ Imposig $S_n=94$, I obtain: $n^2+n-94=0$ and so $n=10$.
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