Question

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

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

Answer 1

The correct option is A $17$

Given, the sequence $1,2,2,3,3,3,4,4,4,4,.....$ where $n$ consecutive terms have the value $n$.

Now, difference of consecutive term are in A.P.

So, $T_n=n\left(\frac{n+1}{2}\right)$

To find ${150}^{th}$ term, put $T_n=150$ in above equation, we have,

$150=n\left(\frac{n+1}{2}\right)$

$\Rightarrow n^2+n-300=0$

$\Rightarrow n=\frac{-1\pm \sqrt{1-4\times 1\times -300}}{2}=\frac{-1\pm \sqrt{1201}}{2}$

$n=16.8$ or $n=-17.8$ (Discarding the negative value).

So, $n=16.8$ and nearest integral value of $n$ is $17$.

Thus, ${150}^{th}$ term of given sequence will be $17$.