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# help

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Erich lists the sequence of non-perfect squares: 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, ...  What is the 1000th term in this sequence?

May 8, 2020

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Since there are 32 perfect squares between 1 and 1032, therefore: 1032 -32 =1,000. So, the 1000th term without the perfect squares will be 1032.

May 8, 2020
#2
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Erich lists the sequence of non-perfect squares:

$$2,\ 3,\ 5,\ 6,\ 7,\ 8,\ 10,\ 11,\ 12,\ 13,\ 14,\ 15,\ 17,\ \ldots$$
What is the 1000th term in this sequence?

Formula: $$a_n = n + \lfloor 0.5 + \sqrt{n} \rfloor$$

$$\begin{array}{|rcll|} \hline \mathbf{a_n} &=& \mathbf{n + \lfloor 0.5 + \sqrt{n} \rfloor } \quad | \quad n=1000 \\\\ a_{1000} &=& 1000 + \lfloor 0.5 + \sqrt{1000} \rfloor \\ a_{1000} &=& 1000 + \lfloor 0.5 + 31.6\ldots \rfloor \\ a_{1000} &=& 1000 + \lfloor 32.1\ldots \rfloor \\ a_{1000} &=& 1000 + 32 \\ \mathbf{a_{1000}} &=& \mathbf{1032} \\ \hline \end{array}$$

May 9, 2020