anemone said:Prove that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$.
Bacterius said:Note that:
$$\sqrt{c + a} < b ~ ~ \iff ~ ~ c + a < b^2 ~ ~ \iff ~ ~ a < b^2 - c$$
For instance we have:
$$\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$$
$$\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}<3$$
$$\sqrt{3+\cdots+\sqrt{2006}}<7$$
$$\cdots$$
Unrolling the nested square roots as above, we must therefore show that:
$$\sqrt{2006} < ((((2^2 - 1)^2 - 2)^2 - 3)^2 - \cdots)^2 - 2005$$
Note that the sequence defined by $a_1 = 2$ and $a_{n + 1} = a_n^2 - n$, corresponding to the sequence of RHS above, is increasing. We prove this by induction by showing that $a_n \geq n + 1$. First $a_1 = 2$ so the base case holds, now assume $a_n \geq n + 1$, then $a_n^2 - n \geq (n + 1)^2 - n$, that is, $a_{n + 1} = a_n^2 - n \geq n^2 + n + 1 \geq (n + 1) + 1$ as $n^2 \geq 1$. By induction, $a_n \geq n + 1$ for all $n$. Thus we can now show that $a_{n + 1} = a_n^2 - n \geq a_n$, as that is equivalent to $a_n^2 - a_n - n \geq 0$, and since $m^2 \geq 2m$ for all $\lvert m \rvert > 1$ it follows that $a_n^2 - a_ n - n \geq a_n - n \geq (n + 1) - n \geq 1 \geq 0$ and so $(a_n)_{n = 1}^\infty$ is increasing. Finally, the sequence becomes greater than $\sqrt{2006} \approx 44.788$ at $n = 4$ by simple arithmetic as $a_4 = 46$, which proves the claim. $\blacksquare$
The statement of the problem is to prove that the expression $\sqrt{1+\sqrt{2+\cdots+\sqrt{2006}}} < 2$ is true.
This problem is significant because it involves a nested radical expression, which is a common type of mathematical problem. It also challenges the concept of infinite nested radicals and their convergence.
The most common approach to solving this problem is by using mathematical induction. This involves proving the statement for a base case, typically $n=2$, and then assuming the statement is true for $n=k$ and proving it for $n=k+1$. This process is repeated until the desired statement is proved for all positive integers $n$.
The number 2006 is not significant in and of itself, but it is commonly used in this problem because it is a large enough number to demonstrate the concept, yet not too large to make the problem overly complex.
This problem has potential applications in various fields such as mathematics, physics, and engineering. It can be used to solve problems involving infinite nested radicals and their convergence, as well as in the development of algorithms for computing nested radicals.