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 |
= |
9n + 9n-1 + ... + 92 + 9 + 1 |
| |
= |
(9n+1 - 1) / (9 - 1) |
| |
= |
(9n - 1)(9n + 1) / 8 |
| |
= |
[(9n - 1)/2][(9n + 1)/2] / 2 |
And the last expression is of the form n(n+1)/2 if we
let n = (9n - 1)/2;