Nested Power Progression

Algebra Level 5

$\large{ \sqrt{2^2+2^3 + 2\sqrt{2^4 + 2^5 + 2^2 \sqrt{2^6+ 2^7 + 2^3 \sqrt{\cdots}}}} } =\, ?$

The answer is 6.

2 solutions

Otto Bretscher
May 30, 2016

Factoring out powers of 2, this simplifies to $2\sqrt{3+2\sqrt{3+2\sqrt{...}}}$ , which comes out to be $\boxed{6}$

But does it converge?

Pi Han Goh - 5 years ago

I usually ask that question, Comrade ;)

We can rely on Comrade Ramanujan's very good work, or we can see for ourselves: The iteration function $f(x)=2\sqrt{3+x}$ , with fixed point 6, is a contraction since $0<f'(x)\leq \frac{1}{\sqrt{3}}$ for non-negative $x$ . Or "draw a cobweb", as I always say...

Otto Bretscher - 5 years ago

Do you know any apps that can draw a cobweb?

Pi Han Goh - 5 years ago

@Pi Han Goh – @Otto Bretscher You should probably see this

Agnishom Chattopadhyay - 5 years ago

@Pi Han Goh – I don't know myself, but our Comrade @Agnishom Chattopadhyay was posting some cool stuff, for example here ... maybe he can help you.

The "cobweb" is not very interesting here since the iteration function $f(x)$ is increasing; it's more like a stair case with smaller and smaller steps converging towards the point $(6,6)$ .

Otto Bretscher - 5 years ago

James Pohadi
May 30, 2016

$\begin{aligned} \sqrt{2^2+2^3 + 2\sqrt{2^4 + 2^5 + 2^2 \sqrt{2^6+ 2^7 + 2^3 \sqrt{\cdots}}}} & =x \\ \sqrt{2^2+2^3 + 2.2\sqrt{2^2 + 2^3 + 2\sqrt{2^4+ 2^5 + 2^2\sqrt{\cdots}}}} & =x \\ \sqrt{2^2+2^3 + 2.2.x} & =x \\ 2^2+2^3 + 2.2.x & =x^2 \\ 4+8+4x & =x^2 \\ x^2-4x-12 & =0 \\ (x-6)(x+2) & =0 \end{aligned}$

$\boxed{x=6}$ $\text{or}$ $x=-2$ $\small \color{#D61F06}{\text{(NA because negative)} }$

Followed the same method

Aditya Kumar - 5 years ago

Nested Power Progression

The answer is 6.

2 solutions

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