APL(Dyalog Unicode), 10 98 bytes SBCS
⊢∊+\⍣2⍤⍳ -1 byte thanks to att.
A tacit function which takes an integer on the right and returns 1 if tetrahedral and 0 if not. Same method as the Vyxal and Uiua answers.
⊢∊+\⍣2⍤⍳ ⍤⍳ # range, then +\ # cumulative sums, ⍣2 # twice ⊢∊ # input is member 💎 Created with the help of Luminespire.