Math CentralQuandaries & Queries


Question from Chinonyerem, a student:

Each of the numbers
1 = 1, 3 = 1+2, 6 = 1+2+3, 10 = 1+2+3+4 ,...
represents the number of dots that can be arranged evenly in an equilateral
                  .        . .
        .         .       . . .             ...
.      . .      . . .    . . . .
This led the ancient Greeks to call a number TRIANGULAR if it is the
sum of consecutive integers, beginning with 1. Prove the following facts
concerning triangular numbers:
(a) A number is triangular if and only if it is of the form n(n+1)/2 for some n >= 1
(b) The integer n is a triangular number if and only if 8n+1 is a perfect square
(c) The sum of any two consecutive triangular numbers is a perfect square
(d) If n is a triangular number, then so are 9n+1, 25n+3, and 49n+6


Try to prove the statement in part (a) by mathematical induction. Parts (b), (c) and (d) follow from (a).


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS