A triangular number T(n) is defined as n(n+1)/2. It is equal to the binomial coefficient (n+12) and is also equal to 0+1+2+⋯+n. When is an integer m≥0 equal to a triangular number T(n) for some integer n≥0? We have n(n+1)=2m. Using the quadratic formula to solve the quadratic equation n2+n−2m=0 for n, we obtain one solution for n:
n=√8m+1−12
The other solution for n is negative (or complex) so we will not use that.
If 8m+1 is not the square of an integer, then n is not an integer. If 8m+1 is the square of an integer, then 8m+1 is odd and thus √8m+1 is odd as well. This means that n=√8m+1−12 is an integer. Thus m is a triangular number if and only if 8m+1 is the square of an integer.
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment