how to work with radicals in algebra. when radical has constant and variable.

Question

I am attempting to understand the concept of mathematical induction but think I am not understanding some of the algebra in the solution.

I can follow the solution up until the part where the radicals are getting simplified.

How do they get from $\sqrt k \sqrt{k+1} + 1$ to $k + 1$ in the last two lines of the algebra of the RHS?

I don't understand how to work with the radicals that have a variable + constant.

Answer 1

It is an inequality. $k+1>k$. Therefore, $\sqrt{k+1}>\sqrt{k}$.

This allows them to write $\sqrt{k}\sqrt{k+1}>\sqrt{k}\sqrt{k}=(\sqrt{k})^2=k$.