Question

IF x=n(n+1) find root x + root x+.........infinity in terms of n.​

Answer 1

Let,

$\begin{aligned}&\sqrt{x+\sqrt{x+\sqrt{x+...\infty}}}=k\\ \\ \Longrightarrow\ \ &x+\sqrt{x+\sqrt{x+...\infty}}=k^2\\ \\ \Longrightarrow\ \ &\sqrt{x+\sqrt{x+...\infty}}=k^2-x\\ \\ \Longrightarrow\ \ &k=k^2-x\\ \\ \Longrightarrow\ \ &x=k^2-k\\ \\ \Longrightarrow\ \ &n(n+1)=(k-1)k\end{aligned}$

Here, both sides are seemed as the product of two consecutive integers. So we can say that n = k - 1 and n + 1 = k.

But being n = k - 1 and n + 1 = k is one possibility, because it can be true that n = - k and n + 1 = 1 - k.

As an example, it is true that 1 × 2 = (-1) × (-2) = 2.

So,

Case 1: Let k = n + 1.

$\begin{aligned}&k=n+1\\ \\ \Longrightarrow\ \ &\sqrt{x+\sqrt{x+\sqrt{x+...\infty}}}=\mathbf{n+1}\end{aligned}$

Case 2: Let k = -n.

$\begin{aligned}&k=-n\\ \\ \Longrightarrow\ \ &\sqrt{x+\sqrt{x+\sqrt{x+...\infty}}}=\mathbf{-n}\end{aligned}$