Question

In the sequence $1,2,2,3,3,3,4,4,4,4,...,$ where $n$ consecutive terms have the value $n$, the ${150}^{th}$ term is

• A. $17$
• B. $16$
• C. $18$
• D. none of these

Answer 1

Answer: (A)

$17$

Let $N=$ Number of times a number $n$ is repeated in the given sequence $\left\{n\right\}$

$\therefore N=1,2,3,\mathrm{4...}$

Thus, sequence $\left\{N\right\}$ is an AP.

Sum of N terms, $S_N=\frac{N(N+1)}{2}$

The ${150}^{th}$ term of sequence $\left\{n\right\}$ will be minimum $N$ such that $S_N\ge 150$

Thus, $\frac{N(N+1)}{2}\ge 150$

$\Rightarrow N(N+1)\ge 300$

Now, ${17}^2=289$ and $17\times 18=306$

$\therefore$ Minimum N for which $\frac{N(N+1)}{2}\ge 150$ is $17$.

Hence, ${150}^{th}$ term will be $17$.