For a natural number \(n\), we call the number \(1+2+3+\cdots + n\) the \(n^{\text{th}}\) triangular number, and we denote it by \(T_n\). Find \(T_n+T_{n-1}\) in terms of \(n\).
For a natural number \(n\), define the \(n^{\text{th}}\) triangular number by \(T_n=1+2+3+\cdots+n\).
Show that \(3T_n+T_{n-1}\) is itself a triangular number, and determine which one.
Among \(12\) identical-looking balls, exactly one has a different weight (we do not know whether it is heavier or lighter than the others).
Using a balance scale, show how to determine the odd ball, and whether it is lighter or heavier, using only three weighings.
You are given \(68\) coins, and all of them have different weights. Using at most \(100\) weighings on a balance scale, find both the heaviest and the lightest coin.