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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)}{

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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

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The smallest b that works is 45^2 = 2025.

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

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

Sep 15, 2020