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A sequence goes 1,2,2,3,3,3,4,4,4,4,....   What is the 254th term?

 Dec 17, 2019
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A sequence goes 1,2,2,3,3,3,4,4,4,4,....   What is the 254th term?

 

\(\begin{array}{|rcll|} \hline \dfrac{(1+n)n}{2} &=& 254 \\\\ n^2+n &=& 508 \\\\ n^2+n-508 &=& 0 \\ n &=& \dfrac{-1 \pm \sqrt{1-4*(-508)}}{2} \\ n &=& \dfrac{-1 \pm \sqrt{1+2032}}{2} \\ n &=& \dfrac{-1 \pm \sqrt{2033}}{2} \\ n &=& \dfrac{-1 + \sqrt{2033}}{2} \\ n &=& \dfrac{-1 + 45.0888012704}{2} \\ n &=& \dfrac{44.0888012704}{2} \\ n &=& 22.0444006352 \\ \hline \end{array}\)

 

The 254th term is 23

 

laugh

 Dec 17, 2019

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