Triangular Numbers in Base 9
Show that every number that consists entirely of 1's when written
in base 9 is triangular.
Recall that the triangular numbers are 1, 3, 6, 10, 15, ...,
with general formula T(n) = n(n+1)/2.
If a number N consists entirely of 1's when written in base 9, then:
N 
= 
9^{n} + 9^{n1} + ... + 9^{2} + 9 + 1 

= 
(9^{n+1}  1) / (9  1) 

= 
(9^{n}  1)(9^{n} + 1) / 8 

= 
[(9^{n}  1)/2][(9^{n} + 1)/2] / 2 
And the last expression is of the form n(n+1)/2 if we
let n = (9^{n}  1)/2;