<s>[INST] What is the 1393th number? What is the 1393th number in this string??

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5........

Could you explain me how to answer this kind of questions .
 [/INST] It will be a bit easier if we start counting at 0.

start  0  1     3        6          10             15
index  0, 1  2  3  4  5  6  7  8  9 10 11 12 12 14 15 16
number 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6,........

 The number 1 starts at index  0 = 0
 The number 2 starts at index  1 = 0+1.
 The number 3 starts at index  3 = 0+1+2
 The number 4 starts at index  6 = 0+1+2+3
 The number 5 starts at index 10 = 0+1+2+3+4
 The number 6 starts at index 15 = 0+1+2+3+4+5
 The number n starts at index      n(n-1)/2.

So we need to find the largest n such that
    n(n-1)/2 ≤ 1392 (because we started counting at 0 instead of 1)
    n(n-1) ≤ 2784

Since  $n(n-1) = 2784$ is about $n^2 = 2784$, a good guess for $n$ would be $\lfloor \sqrt{2784} \rfloor = 52$
We compute
\begin{align}
   52\times 51 &= 2652\\
   53 \times 52 &= 2756\\
   54 \times 53 &= 2862\\
\end{align}
 So  the number 53 starts at index 53(53-1)/2 = 1378
 and the number 54 starts at index 54(54-1)/2 = 1431.

 That means that the 1379th number is 53
 and the 1432th number is 54.

 So the 1393th number is 53.

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