The $$100^{th}$$  term of the sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, . . . . .$$       is

Solution :
$$1^{st}$$ term $$→ 1,$$  $$2^{nd}$$ term $$= 2,$$  $$4^{th}$$ term $$→ 3,$$  $$7^{th}$$ term $$→ 4,$$  $$11^{th}$$ term $$→ 5, . . . . .$$
Series is $$1, 2, 4, 7, 11, . . . . .$$
$${a_n} = 1 + \frac{{n\left( {n - 1} \right)}}{2} = \frac{{{n^2} - n + 2}}{2}$$
If $$n = 14,$$  then $$a_n = 92,$$
If $$n = 15,$$  then $$a_n = 106.$$