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Home Q & A Board Science-Related Homework Help Mathematics
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Anonymous peoplemover
wrote...
A year ago
Demonstrate Gauss' formula for triangular numbers by finding a triangular number that is a multiple of 19.
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Anonymous
wrote...
#1
A year ago
The first 19 triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190

The Gauss formula for triangular numbers is:

\(T_n=\frac{n\left(n-1\right)}{2}\)

Rearranging:

\(2\cdot T_n=n^2-n\)

\(0=n^2-n-2\cdot T_n\)

Substitute \(n=19\) into the equation above, and solve for \(T_n\):

\(0=19^2-19-2\cdot T_n\)

\(2\cdot T_n=380\)

\(T_n=\frac{380}{2}=190\)

Thus, 190 is the first time we obtain a triangular number that is divisible by 19.
Anonymous Author
wrote...
#2
A year ago
Thank you, this was so helpful!  Heavy Heart
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