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Siobhan, Ruth, Ruth and Toby from Risley Lower Grammar Primary School, Joe from Hove Park Lower School, Nathan, Dulan and Pavan from Wilson's Grammar School and Thomas from PS6 in New York noticed that:
Students from the Tower Hamlets Enriching Maths project also worked on this problem:
Jinquan from The Chinese High School in Singapore explained
that:
$T_{250}+T_{250}$ is $250 \times251$, and more generally $T_{n}+T_n
= n(n+1)$
Hence, $T_n=n(n+1)/2$
which gives $T_{250}=31375$,
Consider $4851$.
If $4851$ is a triangular number, $9702$ can be expressed as $n(n+1)$.
By solving the quadratic equation or by estimating, we get $n = 98$, and hence $4851 = T_{98}$
In general, a number $x$ is a triangular number if and only if $n(n+1)=2x$ is solvable for positive integers of $n$.
Consider $6214$.
If it is a triangular number, $12428$ can be expressed as $n(n+1)$.
By solving the quadratic equation or by estimating, we see that there are no solutions in positive integers.
Hence $6214$ is not a triangular number.
Consider $3655$.
If it is a triangular number, $7310$ can be expressed as $n(n+1)$.
By solving the quadratic equation or by estimating, we get $n = 85$, and hence $3655 = T_{85}$