Math

A triangular number is a number that can be represented by dots arranged in a triangular shape as shown below. The first four triangular numbers are 1,3,6,10. What is the 10th triangular number? the 20th? the 100th?
1st 2nd 3rd 4th
. . . .
. . . . . . . . . . . .
. . . .
Is there a simple formula for this?


0, 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, 1326, 1378, 1431

Search on Trianglular numbers on Google.


[k(k+1)]/2 is the general formula.
k=1 gives (1*2)/2=1
k=2 gives (2*3)/2=3
k=10 (10*11)/2 = something you can do.
k=20 (20*21)/2 = something you can do.
k=100 (100*101)/2 = something you can do.
Prove the formula by induction.



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