Challenge #2

Posted on March 10, 2006 by Brent

Can you apply the techniques learned from the solution to Challenge #1 in order to find a value for these expressions?

1. $\displaystyle 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}$

2. $\displaystyle 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{3 + \ddots}}}$

3. $\displaystyle \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}}}$

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

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Elissa says:

March 16, 2006 at 4:48 pm

1) 2.5
2) 1.847
3) 1.618

i think…

and, just so you know…I did these while procrastinating from studying for my calc test tomorrow…
Brent says:

March 16, 2006 at 5:34 pm

#3 is right, but #1 and #2 are not. guess you have some more procrastination ahead of you… (=
Elissa says:

March 17, 2006 at 2:33 pm

hmmmm….well, procrastination is no longer needed…but fun is!

1) 2.414 I am not sure WHAT I was thinking the first time…I did some very dumb things….oh, silly me!

2) 1.333…
Brent says:

March 17, 2006 at 4:41 pm

OK, you got #1 now. sorry to hear about the dumb things. (= #2 is still not right. How are you trying to solve it?
Gordo says:

March 18, 2006 at 5:45 pm

Is #2 1.264?
Brent says:

March 19, 2006 at 3:04 pm

yes, that’s right. Now, to a real mathematician, none of these answers would be technically correct since they are all approximations (the answer to #2 is not exactly 1.264, and so on). Does anyone know the exact answers (in terms of square roots and so on)?
Gordo says:

March 19, 2006 at 4:20 pm

Ohh…this is fun. I guess it would be (3+√(21))/6?
Brent says:

March 19, 2006 at 4:23 pm

you got it. Glad it is fun!
sulgi says:

March 27, 2006 at 12:50 pm

brentos… hallo! i must confess i didnt try any of the math problems, but i just want to say hi to you and to mrs. joyia yorgey as well… hehe that still feels weird to say!