I need a Math Guru for this one! Ok?

Question

What does 1/1+1/1+1/1+1/1+1....Equal? In other words One Divided by One Plus One Divided by One Plus One Divided...It has something to do with Phi.

My second one is similar:
The Square Root of One Plus The Square Root of One Plus The Square Root of One...Equals.

If any smart Math Guru has the Answer, I'd really appreciate it.

Answer 1

If you precede this with a 1,

1 + 1/(1 + 1/(1 + 1/(1 + ... ) ) )

then you get the golden ratio, Phi.

The second one is also equal to Phi.

Let x = sqrt(1 + sqrt(1 + sqrt(1 + ...)))

Then x^2 = 1 + x. The positive root of this is Phi.

Answer 2

I think I know what you mean...

1/
1+1/
1+1/
1+1/
...
clearly it is less than 1/1 = 1 and is more than 1/(1+1) = 1/2

the 2nd as well
rt(1+rt(1+rt(1+rt(1+...

again we see it is clearly more than rt1 = 1 and clearly less than rt(1+2) = rt(3) which is roughly 1.732

throughout the process I will call the square root simply "rt"

for the 1st one...
let 1/1+1/1+... = x
now then lets add 1 to both sides
1 + 1/1+1/1+... = x+1
notice anything? now flip it over
1/ 1+ 1/1+1/... = 1/(x+1)
But THEN you get what you started with so
x = 1/(x+1)
x^2 + x = 1
x^2 + x - 1 = 0
complete the square
x^2 + x + 1/4 - 5/4 = 0
(x+1/2)^2 - (rt5/2)^2 = 0
then if A^2 - B^2 = 0
A+B = 0 or A-B = 0
thus x is -1/2 plus or minus (rt5)/2
but x is CLEARLY a positive number
thus x = ((rt5)-1)/2
which is about 0.618
which seems logical considering our first observations
one problem finished!

prob no.2 again, but just to prevent confusion, I'm calling it y

y = rt(1+rt(1+...
again add 1 to both sides
y+1 = 1+rt(1+rt(1+...
now take the rt of both sides
rt(y+1) = rt(1+rt(1+...
but again this puts us back to the start so...
rt(y+1) = y
square both sides
y+1 = y^2
y^2 - y - 1 = 0
complete the square
y^2 - y +1/4 - 5/4 = 0
(y-1/2)^2 - ((rt5)/2)^2 = 0
in the same way we conclude that
y = 1/2 plus or minus (rt5)/2
y is clearly positive (and must be such, since it is a root!)
thus y = (1 + rt5)/2
which comes out about 1.618
consistent again!

SOLVED!

signed,
Math guru :p

Answer 3

As written, because you do division before addition, the expression is

(1/1) + (1/1) + (1/1) + ... = 1 + 1 + 1 + ...

I believe you mean

1 / ( 1 + 1 / ( 1 + 1 / ( 1 + ... ) ) ) )

If you let that equal x, then

x =1 / ( 1 + x ), or

x ( 1 + x ) = 1, or

x + x² = 1, or

x² + x - 1 = 0

with solution x = [ -1 ± √(1 + 4 ) ] / 2 = ( -1 ± √5 ) / 2

Because x is positive, the answer is (√5 - 1 ) / 2 = 0.618... = φ

Similarly, let

y = √( 1 + √( 1 + √ ( 1 + ... ))), then

y² = 1 + y, or

y² - y - 1 = 0

with solutions

y = [ 1 ± √(1 + 4 ) ] / 2 = ( 1 ± √5 ) / 2

Take the positive solution:

y = (√5 + 1 ) / 2 = 1 / φ = 1.618...

Answer 4

It is an infinity of order Aleph = 0. It is equal to the cardinality of the set of natural numbers. Between two natural numbers there is an infinity of rational numbers of the form A/B where A and B are natural numbers. So the cardinality of the set of rational number is Aleph = 1. Between two rational number there is an infinity of irrational numbers. Numbers with infinite and aperiodic decimal representation. Like Pi or the square root of 2. So the cardinality of the set of real number is Aleph = 2.
There must be a parenthesis error in your statement
you must mean
1 + 1/(1+1)+ 1/(1+1+1) ... = 1 +1/2 + 1/3..
or something similar

Answer 5

Something is wrong, because 1/1 is 1 so 1/1 + 1/1 etc is the same as 1+1+1+1... which of course is infinity.

Same with square root. I think you mean something different...

Answer 6

i dont actually understand what u have wrote in here. bt a great time before greeks and romans(archimedes and the rest) used this kind of recursive formuls to approximate some numbers like phi and the golden ratio(1.6 something). and they did it like in that long devision pattern like

1+ 1/(1+1/(1+1/ and so on

i havent actually tried to simplify it or prove it but i think it comes to someware around phi/2. check it out in some math history book or a site.

Answer 7

easy ( you mean continued fraction)

let 1+1/1+1/+ ... be x

then we have 1+1/(1+1/+1 ... ) = x
or 1+1/x = x

this is same as phi.

now you can work out

second one is similarly

sqrt(1+1/sqrt(x)) = x
you should be able to proceed

Answer 8

I believe they are trick questions to see who is paying attention
1/1 and the square root of one are both = to 1
you might be reading too deep into the
question when you don't need to.

Answer 9

It depends on how many (1/1)s you have, the answer is

n, whereas n stands for the number of (1/1)s you have

same answer from sqrt(1) + sqrt(1) + sqrt(1) + ....

it depends on how many values you have.

Answer 10

You don't really need to be a math guru to answer these two.