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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
 #1
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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: \(a_n = n + \lfloor 0.5 + \sqrt{n} \rfloor \)

 

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

 

see: https://oeis.org/search?q=2%2C+3%2C+5%2C+6%2C+7%2C+8%2C+10%2C+11%2C+12%2C+13%2C+14%2C+15%2C+17&language=english&go=Search

 

laugh

 May 9, 2020

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