Question

Let the positive integers be written in the form :

If the $k^\text{th}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number $5310$ will be, is _____

Answer 1

The total number of elements in the first n rows is:
$S = 1 + 2 + 3 + \dots + T_n = \frac{n(n+1)}{2}.$
To find the row containing 5310, solve:
$\frac{n(n+1)}{2} = 5310.$
Start testing values:
$n = 100, \quad T_n = \frac{100 \cdot 101}{2} = 5050.$
$n = 101, \quad T_n = \frac{101 \cdot 102}{2} = 5151.$
$n = 102, \quad T_n = \frac{102 \cdot 103}{2} = 5253.$
$n = 103, \quad T_n = \frac{103 \cdot 104}{2} = 5356.$
Since 5310 lies between 5253 and 5356, it is in the 103rd row.
Final Answer: 103.