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For a positive integer \(n\), the \(n^{th}\) triangular number is \(T(n)=\dfrac{n(n+1)}{2}.\)

For example, \(T(3) = \frac{3(3+1)}{2}= \frac{3(4)}{2}=6\), so the third triangular number is 6.

Determine the smallest integer \(b>2011\) such that \(T(b+1)-T(b)=T(x)\) for some positive integer .

 Sep 15, 2020
 #1
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The smallest b that works is 45^2 = 2025.

 Sep 15, 2020
 #2
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T[2016]  -  T[2015]  = 2016 =T[63]

 

T[63] =[63 x 64] / 2 = 2016

 Sep 15, 2020

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