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For a positive integer n , the nth  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 x.

 Jul 13, 2022
 #1
avatar+124525 
+1

T(b + 1)  - T(b)  =  

 

(b + 1) (b + 2)  - (b) (b + 1)

______________________   =

                   2

 

(b + 1)  ( b + 2 - b)

_______________     =  

                 2

 

2 ( b + 1)

_______  =   

      2

 

b +  1

 

T(63)  = 2016

 

So

 

b + 1 =   T(63)

 

b + 1 =  2016

 

b  =  2015

 

cool cool cool

 Jul 13, 2022

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