Hi Vedhanthika,
Given that a > b, we can write it as a $$\ge\ $$ b + 1. Similarly we obtained $$10^{\dfrac{n\left(n+1\right)}{2}}>999$$ we can write it as $$10^{\dfrac{n\left(n+1\right)}{2}}\ge\ 1000$$ and $$\dfrac{n\left(n+1\right)}{2}\ge\ 3$$. 
From here, we obtained the minimum value of n as 6.
Hope this helps!

Hi,
i do not understand how did you get value 3 in last step

Hi Bhanu,
We had the equation,
$$10^{\dfrac{n\left(n+1\right)}{2}}\ge\ 1000$$
This can also be written as,
$$10^{\dfrac{n\left(n+1\right)}{2}}\ge\ 10^3$$
Since both the bases are the same positive real numbers, we can directly compare the numbers in the power, and by comparing the powers, we obtain the condition that the power in LHS has to be greater than the power in RHS,
$$\dfrac{n\left(n+1\right)}{2}\ge\ 3$$
Hope this helps!