ISI2016-DCG-9

Question

The $5000$th  term of the sequence $1,2,2,3,3,3,4,4,4,4,\cdots$ is

• A. $98$
• B. $99$
• C. $100$
• D. $101$

Answer 1

Answer: C

Explanation:

Small observation:

Every number is repeated by the same number of times.

Eg. $1$ repeats $1$ time, $2$ repeat $2$ times, $\cdots$

Now, this question reduces to nothing but just a sum of n terms of natural numbers, which is given by:

$\frac{n(n-1)}{2} < 5000 \le \frac{n(n+1)}{2}$

On substituting the options, we will get that 100 satisfies this equation.

$\therefore \mathbf C $ is the right option.

Comments

I think the correct inequality will be $\frac{n(n-1)}{2}<5000\le \frac{n(n+1)}{2}$

[representing the position of $n$ in the sequence] where $n$ is the required repeating number.

---

@`JEET

$\frac{n(n+1)}{2}\ge5000\Rightarrow n\ge100 \Rightarrow n=100,101,102,...$

The inequality above doesn't produce the unique answer.

 

But this inequality $\frac{n(n-1)}{2}<5000\le \frac{n(n+1)}{2}$ produce only $n=100$.

---

I think for objective technique that is correct!

---

I have taken the minimum value, it can satisfy even more than $2$ values as well. Don't you think its correct keeping in view of objective problem solving?

But I still change to make it mathematically correct.