Question

please answer I will mark as brain list and I will like it ​

Answer 1

$\large\text{\underline{Short note:-}}$

This problem was introduced in a video of BPRP. I recommend you to watch his video if you want to understand more.

Also, your question doesn't include mode value, please check if the question is incorrect.

$\large\text{\underline{Solution:-}}$

First, let's calculate $\sqrt{a(a+1)(a+2)(a+3)+1}$.

If we make the coefficient equal, it is simple to simplify.

$\text{(Given expression)}$

$=\sqrt{(a^{2}+3a)(a^{2}+3a+2)+1}$

$=\sqrt{(a^{2}+3a)^{2}+2(a^{2}+3a)+1}$

$=|a^{2}+3a+1|$

Using the previous answer with $a=50$,

$\sqrt{50\cdot51\cdot52\cdot53+1}$

$=|50^{2}+3\times50+1|$

$=|2500+150+1|$

$=|2651|$

$=2651$

$\large\text{\underline{Conclusion:-}}$

So, the correct option is choice 2) $2651$ & $|a^{2}+3a+1|$.