a function $f$ satisfies the following conditions:
$$\begin{align*}
&f(1)=1\\
&f(n)=f(n-1)+2\sqrt{f(n-1)}+1\quad\text{for integers}\quad n\ge 2
\end{align*}$$
Find a formula that might be true for all integers $n\ge1$. Then prove using mathematical induction that it is indeed correct.
I found that the formula is $f(n)=n^2$, but I'm stuck on how to prove it... can anyone help me please?
Thanks a lot!
Answer
HINT for the induction step: if $f(n)=n^2$, then $$f(n+1)=f(n)+2\sqrt{f(n)}-1=n^2+2\sqrt{n^2}+1=\dots~?$$
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