Unit 2 · Topic 10 · Number Play
Pixel had already shown that 6 and 10 — two consecutive triangular numbers — add up to 16, a perfect square.
Kabir wanted to know: which square, for which pair, without checking every single one by hand.
Anaya wrote a single short formula that answered it for every pair, all at once.
"We already know 6 + 10 = 16," said Pixel, "and that 16 is 4 squared. But which two triangular numbers give you 10 squared, without checking pair after pair?" Kabir started listing triangular numbers by hand: 1, 3, 6, 10, 15, 21, 28, 36, 45 — planning to add consecutive pairs one at a time until he found 100.
Anaya stopped him. "There's a shortcut. The nth triangular number is n(n+1)/2. The (n+1)th triangular number is (n+1)(n+2)/2. If you add those two general expressions together and simplify, everything collapses down to (n+1) squared."
Kabir worked through it slowly with her: n(n+1)/2 + (n+1)(n+2)/2. "Factor out (n+1) from both," said Anaya, "and you get (n+1) x [n/2 + (n+2)/2], which simplifies to (n+1) x [(n + n + 2)/2] = (n+1) x [(2n+2)/2] = (n+1) x (n+1)." "That's just (n+1) squared," said Kabir.
"Exactly," said Pixel. "So for the pair (n, n+1), meaning the nth and (n+1)th triangular numbers, the sum is always (n+1) squared — no matter which n you pick." Kabir tested it directly: for n=3, the 3rd triangular number is 6, the 4th is 10, and 6+10=16, which is indeed (3+1) squared = 16. It matched exactly.
"So to find the pair that sums to 100," said Anaya, "I need (n+1) squared = 100, so n+1 = 10, meaning n = 9." The 9th triangular number is 45, and the 10th is 55. "45 + 55 = 100 — exactly right, and I never had to list a single triangular number by hand to find it."
The nth triangular number is given by the formula n(n+1)/2. Adding two CONSECUTIVE triangular numbers — the nth and the (n+1)th — always gives a perfect square: specifically, (n+1) squared.
This nth-term formula, (n+1) squared, lets you predict the sum of any pair of consecutive triangular numbers instantly, without listing or adding the triangular numbers one by one.
The formula also works in reverse: to find WHICH pair of consecutive triangular numbers sums to a given square number, take the square root of that number, subtract 1, and that gives you n — the position of the first triangular number in the pair.
Use the formula to find the sum of the 5th and 6th triangular numbers without listing them, then check by adding the actual triangular numbers.
Find the pair of consecutive triangular numbers that sum to 49.
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