JEE Main 2024 — Sequences And Series Question with Solution

To solve this problem, we need to determine in which row the number $5310$ appears when positive integers are arranged in rows such that the $k^{\text{th}}$ row contains exactly $k$ numbers.

Understanding the Pattern

First row ($k=1$): Contains 1 number.

Second row ($k=2$): Contains 2 numbers.

Third row ($k=3$): Contains 3 numbers.

…

$k^{\text{th}}$ row: Contains $k$ numbers.

Total Numbers Up to the $k^{\text{th}}$ Row

The total number of integers up to the $k^{\text{th}}$ row is given by the sum of the first $k$ natural numbers:

$S(k)=1+2+3+\cdots +k=\frac{k(k+1)}{2}$

Finding the Row Containing 5310

We need to find the smallest integer $k$ such that $S(k-1)<5310\le S(k)$. This means:

$\frac{(k-1)k}{2}<5310\le \frac{k(k+1)}{2}$

Step 1: Estimate $k$

Let's approximate $k$ by solving the inequality:

$\frac{k(k+1)}{2}\ge 5310$

Multiply both sides by 2:

$k(k+1)\ge 10620$

This is a quadratic inequality. We can approximate $k$ by taking the square root:

$k\approx \sqrt{10620}\approx 103$

Step 2: Calculate $S(k)$ for $k=102$ and $k=103$

For $k=102$:

$S(102)=\frac{102\times 103}{2}=\frac{10506}{2}=5253$

For $k=103$:

$S(103)=\frac{103\times 104}{2}=\frac{10712}{2}=5356$

Step 3: Determine the Correct Row

Since $S(102)=5253<5310\le 5356=S(103)$, the number $5310$ lies between $5254$ and $5356$, inclusive. Therefore, it is in the $103^{\text{rd}}$ row.

Conclusion

The number $5310$ will be in the $103^{\text{rd}}$ row.