Consider the sequence $1,2,2,3,3,3,4,4,4,4, \ldots$ where $n$ consecutive terms have the value n ..

Question

Consider the sequence $1,2,2,3,3,3,4,4,4,4, \ldots$ where $n$ consecutive terms have the value n .. Then the $150^{	ext {th }}$ term is

Professor PC Thomas Classes · Answer

Let \  $\mathrm{t}_{150}=\mathrm{n}$ $ \begin{gathered} 1+2+3+\ldots(\mathrm{n}-1)<150<1+2+3 \ldots \mathrm{n} \ \mathrm{n}(\mathrm{n}-1)<300<\mathrm{n}(\mathrm{n}+1) \ \Rightarrow \mathrm{n}=17 \end{gathered} $