Question

Diophantus, a Greek mathematician who lived in the 3rd Century B.C., found a connection between triangular numbers (T) and square numbers (K) Find the square number when the triangular number is

a). 6
b). 21
c). 78
d). 276
e). 406
f). 820

Answer 1

The connection between triangular numbers and square numbers established by Diophantus can be understood by finding the ordinal position of the triangular number and squaring it to get the square number. For the given triangular numbers 6, 21, 78, 276, 406, and 820, the corresponding square numbers are 9, 36, 144, 529, 784, and 1600 respectively.

Step-by-step explanation:

Diophantus, a Greek mathematician, is known for his work in algebra and has been credited with finding connections between triangular numbers (T) and square numbers (K). A triangular number is a number that can form an equilateral triangle, and it is represented by the formula Tn = n(n+1)/2, where n is a positive integer. Diophantus's findings relate to the fact that the square of the triangular number can be visualized in terms of dots forming a perfect square.

To find the square number when given a triangular number, we simply take the square root of the triangular number and then square that result. Let's apply this method to find the square numbers for the provided triangular numbers

1. For T = 6, which is the 3rd triangular number (since 3(3+1)/2 = 6), the corresponding square number is K = 32 = 9.
2. For T = 21, which is the 6th triangular number (since 6(6+1)/2 = 21), the corresponding square number is K = 62 = 36.
3. For T = 78, which is the 12th triangular number (since 12(12+1)/2 = 78), the corresponding square number is K = 122 = 144.
4. For T = 276, which is the 23rd triangular number (since 23(23+1)/2 = 276), the corresponding square number is K = 232 = 529.
5. For T = 406, which is the 28th triangular number (since 28(28+1)/2 = 406), the corresponding square number is K = 282 = 784.
6. For T = 820, which is the 40th triangular number (since 40(40+1)/2 = 820), the corresponding square number is K = 402 = 1600.

This illustrates the connection established by Diophantus between triangular numbers and square numbers.