Question 17

Number Theory Level pending

For a positive integer n , n, the n t h n^{th} triangular number is T ( n ) = n ( n + 1 ) 2 . T(n)=\frac{n(n+1)}{2}. For example, T ( 3 ) = 3 × 4 2 = 6 , T(3)=\frac{3\times4}{2}=6, so the third triangular number is 6. 6.

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


The answer is 2015.

Trevor B. by Trevor B. , 22, USA

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1 solution

Hatim Zaghloul
Jun 5, 2014

T(b+1)-T(b) = (b+2)(b+1)/2 -(b+1)(b)/2 = b+1. Hence, we are looking for a number greater than 2011 that is a rectangular number. The square root of 2011*2 is 63.4. The triangular number of 63 is 2016. So, b+1 = 2016. b=2015.

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