Which is the next triangular number in the sequence: 1, 3, 6, 10, 15, 21, .......?
Triangular number formula. Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers. Every even perfect number is triangular (as well as hexagonal), given by the formula where is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are triangular. For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9.
Triangular number Hence every triangular number is either divisible by three or has a remainder of 1 when divided by 9: The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9". The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three. If is a triangular number, then is also a triangular number, given is an odd square and The first several pairs of this form (not counting )