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
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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.
HELP MATH
To find the 10th triangular number, you can use the formula for triangular numbers: [k(k+1)]/2.
Substitute k = 10 into the formula: (10 * 11) / 2 = 55.
Therefore, the 10th triangular number is 55.
To find the 20th triangular number, substitute k = 20 into the formula: (20 * 21) / 2 = 210.
Therefore, the 20th triangular number is 210.
To find the 100th triangular number, substitute k = 100 into the formula: (100 * 101) / 2 = 5050.
Therefore, the 100th triangular number is 5050.
You can further verify the formula by using the induction method, which involves proving that the formula holds true for the base case (k = 1) and then proving that if it holds true for some value of k, it also holds true for the next value of k. This method validates the formula for all positive integers.
To learn more about triangular numbers and derive the formula using induction, you can search for "Triangular numbers" or "Derivation of triangular number formula" on Google.