NIMCET 2015 — Mathematics PYQ

Looking at the pattern of the sum, we can write the $n^{th}$ term ($T_n$) as:

where $n$ goes from $1$ to $24$.

2. Rationalize the General Term

To simplify $T_n$, we multiply the numerator and the denominator by the conjugate $(n+1)\sqrt{n}-n\sqrt{n+1}$:

The denominator is in the form $(a+b)(a-b)=a^2-b^2$:

So, the simplified general term is:

3. Split the Term

Now, divide each part of the numerator by the denominator:

4. Calculate the Total Sum

This is a telescoping series. Let's sum $T_n$ from $n=1$ to $n=24$:

Expanding the terms:

All intermediate terms cancel out, leaving only the first and the last term:

Correct Option: (b)