Mega A Level Maths Thread MK II

You already solved this on post 3186


Look at the underbraced bit

LOL, never type like that again. ever.

I got y=1+root 5 /2 from completing the square, please tell me I'm right

then $\dfrac{x}{\sqrt 10}=\dfrac{1+\sqrt 5}{2}$

therefore $x=\dfrac{\sqrt 10 + 5\sqrt 2}{2}$

Yes, but that doesn't mean I understand the concept It's all equal to 1 at the end?

I got y=1+root 5 /2 from completing the square, please tell me I'm right

then $\dfrac{x}{\sqrt 10}=\dfrac{1+\sqrt 5}{2}$

therefore $x=\dfrac{\sqrt 10 + 5\sqrt 2}{2}$

Correct

Felix this if for you This is a problem by Ramanujan so it is hard.

Can you explain what phi is to me, apparently it has some importance here, and you constantly used it :P

Woah, one at a time guys. More than enough Felix to go around


No xD Ok, you have $y = \sqrt{1 + \sqrt{1 + \sqrt{1 + ... }}} \Rightarrow y^{2} = 1 + \displaystyle\underbrace{\sqrt{1 + \sqrt{1 + \sqrt{1 + ... }}}}_{\alpha}$

Does $\alpha$ look familiar?

Whatz wrong wit tlkin lyk dis Aha just kidding I'd never be caught dead talking like that in real life And I can only imagine how Felix will reply to this, if he does that is I wouldn't say I'm clever though as on TSR I'm pretty average really

I'll get back to you.

No problem. But can you explain the nested sqrt1 thingy?

$x = \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} \Rightarrow x^{2} = 1 + \displaystyle\underbrace{\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}}_{x} \Leftrightarrow x^{2} -x -1 = 0$

$\Rightarrow x = \dfrac{1 + \sqrt{5}}{2}$ (we take the positive square root as $x > 0$)

Oh sorry I understand this but what I can't understand is how is that possible when the sqrt1 is simply 1?

Haha, 'in my ends' you'd be surprised at how many people talk like that >.<
At school everyones like 'help me with this, how do you do that?' etc whereas on TSR, I feel so clueless, more like me asking for help!

$\phi = \dfrac{1 + \sqrt{5}}{2}$

It's just a number, like pi and it's just as important because it appears everywhere It's called the golden ratio if you're interested - you could think of it as being the positive root of $x^{2} -x-1=0$ - I kept using it for these questions because this equation keeps popping up a lot for these questions, hence phi keeps popping up a lot

But you're not taking just square root 1 You have repeating radicals here. Try testing it for small values of a finite series rather than the infinite...

$\sqrt{1 + \sqrt{1 + \sqrt{1}}} = \sqrt{1 + \sqrt{1 + 1}} = \sqrt{1 + \sqrt{2}}$

if you want a proof that it converges, then you may need to give me a bit more time.

Thanks! One thing, do I have to write down that annoying greek letter for phi, because I'm finding it literally impossible to write down neatly, would rather do that theta like thing

Seriously? I would have thought that it would be one of the easier greek letters to write down

I occasionally have trouble writing uppercase sigma neatly

I find greek letters elegant and especially sigma

I find greek letters elegant and especially sigma

See I like sigma! there's a reason I have a laptop for the rest of my exams apart from maths

Seriously? I would have thought that it would be one of the easier greek letters to write down

I occasionally have trouble writing uppercase sigma neatly

I can't write lowercase sigma as it ends up looking like a 6 o.O