prove-it-times-n-4-4-4-4-lt-3-

Question Number 192062 by mehdee42 last updated on 07/May/23

p r o v e i t :

t i m e s_n; \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots + \sqrt{4}}}} < 3

Commented by ajfour last updated on 08/May/23

b u t 4 > 3

h o w c a n t h i s b e \dots

r e a l l y n o u s e a r g u i n g!

Commented by ajfour last updated on 08/May/23

sometimes it just really dont make no sense et al.

Commented by mehdee42 last updated on 08/May/23

p a y a t t e n t i o n t o q u e s t i o n

t h a t i s (\sqrt{4} < 3) r i g h t, n o t w h a t y o u w r o t e (4 > 3)

Answered by Frix last updated on 07/May/23

x = \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots}}} > 0

x = \sqrt{4 + x}

x^{2} - x - 4 = 0

x = \frac{1 + \sqrt{17}}{2}

\frac{1 + \sqrt{17}}{2} < 3

1 + \sqrt{17} < 6

\sqrt{17} < 5

17 < 25 true

Commented by mehdee42 last updated on 07/May/23

p a y a t t e n t i o n :

\sqrt{4 + \sqrt{4 + \sqrt{4 + \dots .}}} \neq \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots \sqrt{4}}}}

\Rightarrow i f x = \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots \sqrt{4}}}} ⇏ x^{2} = 4 + x

t h e e x p r e s s i o n o n t h e l e f t c o n t a i n s t h e i n f i n i t i v e o f t h e

s e n t e n c e . w h i l e t h e n u m b e r o f s e n t e n e s

i n t h r r i g h t e x p e r e s s i o n i s f i n i t .

Commented by Frix last updated on 07/May/23

x = \frac{1 + \sqrt{17}}{2}

\sqrt{4} < \sqrt{4 + \sqrt{4}} < \sqrt{4 + \sqrt{4 + \sqrt{4}}} < \dots < x < 3

Commented by mehdee42 last updated on 07/May/23

w h y “ \dots < x < 3^{″} ?!

Commented by AST last updated on 07/May/23

L e t 4_{n} = \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots + \sqrt{4}}}} (w h e r e 4 a p p e a r s n

t i m e s)

4_{n} < 4_{p} w h e n n < p \dots (i)

T h i s i s t r u e s i n c e 4_{n} = \sqrt{4 + z} a n d z i n c r e a s e s a s

n i n c r e a s e s .

s i n c e 4_{\infty} < 3, (i) \Rightarrow 4_{n} < 4_{\infty} < 3

H e n c e, w e h a v e s h o w n t h a t f o r a l l n, 4_{n} < 3 .

Commented by mehdee42 last updated on 07/May/23

s i r

w h y 4_{n} < 4_{\infty} < 3 ? ?

Commented by mehdee42 last updated on 07/May/23

i t c a n b e p r o v e n b y a v e r y s i m p l e

m a h e m a t i c a l i n d u c t i o n m e t o d

g o o d l u c k

Answered by mehdee42 last updated on 08/May/23

a n s w e r t o q u e s t i o n n u m b e r

l e t : p_{n} = n_t e r m e s \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots + \sqrt{4}}}} < 3

p_{1} = \sqrt{4} = 2 < 3 ✓ i . s

p_{k} = (k_t e r m e s) \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots . + \sqrt{4}}}} < 3 i . h

p_{k + 1} = (k + 1_t e r m e s) \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots + \sqrt{4}}}} < 3 ? i . r

p_{k + 1}^{2} = 4 + (k_t e r m e s) \sqrt{4 + \sqrt{4 + \sqrt{4 + \dots + \sqrt{4}}}} < 4 + 3 = 7

\Rightarrow p_{k + 1} < \sqrt{7} < 3 ✓

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