When I was reading Feynman lectures , I encountered with 'random walk' concept in which he states the expected distance to be square root of N where N is number of steps but, when I searched online for another solution the answer was given like this:-
Then, which is the correct answer and why?
I am attaching solution of both wolfram alpha and feynmann lectures.
enter link description here
I don't think the answers are different, you can see the wolfram alpha answer matches exactly the Feynman one: 
The formula you took a screenshot of is for the expected value of the absolute distance from the the starting location, where as the root mean square distance described earlier is the square root of the expected value of the square of the distance from the origin. For example, imagine you sample 3 different distances, -1, 2, and 3. the average absolute value of the distances is (|-1|+|2|+|3|)/3=(1+2+3)/3=2 but the root means squared value is the square root of the average distances squared which means it is sqrt(((-1)^2+(2)^2+(3)^2)/3)=sqrt((1+4+9)/3)=sqrt(14/3)=2.16 which as you can see is not the same 2, the average absolute distance.
As to why these different ways of conceptualizing the distance exists is a giant rabbit hole I wouldn't waste your time with unless you are genuinely interested to learn more. But my limited understanding of things is basically that math and statistics are invented by humans, so we can make whatever distance measures we want, and maybe in some cases, some measures will be more useful than others.
Final note: Keep in mind that if you just wanted to measure the average distance from the origin after N steps, you'd get an answer of zero, because half of your distances would be far to the left, and the other half far to the right, and they'd cancel out. Generally people want to avoid this, so they either measure the absolute value of the distance or the square of the distance (remember -1*-1 = +1).
