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If the numbers on the third diagonal of Pascal's Triangle are triangular numbers, what is the value of the $50$th triangular number? (The $n$th triangular number is $1+2+3+\ldots+n$.)

 Aug 5, 2020
 #1
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Triangular numbers begin like this:

 

(1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275)  - And 1275 is the 50th triangular number.

 Aug 5, 2020
 #2
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There is a formula to find the nth term of a triangular number:  tn  =  n(n + 1)/2

So, as Guest as shown, the 50th term is:  50(50 + 1)/2  =  1275.

 Aug 5, 2020

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