_{1}(n) be the n-th Triangular number, the sum of 1 + 2 + ... + n.

T_{2}(n) is the sum of the first n Triangular numbers.

T_{3}(n) is the sum of the first n T_{2}(n) numbers.

...

T_{k}(n) is the sum of the first n T_{k-1}(n) numbers.

Please derive closed form expressions for each of T_{1}(n), T_{2}(n), and T_{3}(n). Describe the technique you used for the derivation.

Bonus: what is the formula for T_{k}(n)?

Bonus 2: prove the validity of your formula for T_{k}(n).