- #1

Pirang

- 10

- 0

Hello, I was not sure if this belonged in the precalculus or calculus section so i hope no one minds that i posted it here.

This is a problem that has completely lost me.

http://img76.imageshack.us/img76/9191/continuousfractionrg3.gif

T1 = 1 + 1

T1 = 1+ (1/1+1)

T3 = 1+ (1/(1+1/1+1)

and so on

I have been able to draw a generalized formula for Tn+1 in terms of Tn and that is:

[PLAIN]http://img76.imageshack.us/img76/3978/continuousfraction2fg8.gif

i realized that as n increases to infinity, (Tn+1) - Tn converges towards 0

There i wrote Tn = Tn+1

and inserting this into

[PLAIN]http://img76.imageshack.us/img76/3978/continuousfraction2fg8.gif

you get

http://img76.imageshack.us/img76/1003/continuousfraction3nu9.gif

Therefore

An exact value for the continued fraction will be considering Tn = x :

x^2 - x - 1 = 0

so the exact value is:

http://img76.imageshack.us/img76/3245/continuousfraction4tw6.gif

Now,

Considering

http://img161.imageshack.us/img161/691/continuousfraction5az5.gif

For any value of k, I am supposed to determine a generalized statement for the exact value of any such continued fraction. For which values of k does this hold true and how do i know?

Can anyone please help me with this, i got so far and do not want to give up. Any tips? anything.

This is a problem that has completely lost me.

http://img76.imageshack.us/img76/9191/continuousfractionrg3.gif

T1 = 1 + 1

T1 = 1+ (1/1+1)

T3 = 1+ (1/(1+1/1+1)

and so on

I have been able to draw a generalized formula for Tn+1 in terms of Tn and that is:

[PLAIN]http://img76.imageshack.us/img76/3978/continuousfraction2fg8.gif

i realized that as n increases to infinity, (Tn+1) - Tn converges towards 0

There i wrote Tn = Tn+1

and inserting this into

[PLAIN]http://img76.imageshack.us/img76/3978/continuousfraction2fg8.gif

you get

http://img76.imageshack.us/img76/1003/continuousfraction3nu9.gif

Therefore

An exact value for the continued fraction will be considering Tn = x :

x^2 - x - 1 = 0

so the exact value is:

http://img76.imageshack.us/img76/3245/continuousfraction4tw6.gif

Now,

Considering

http://img161.imageshack.us/img161/691/continuousfraction5az5.gif

For any value of k, I am supposed to determine a generalized statement for the exact value of any such continued fraction. For which values of k does this hold true and how do i know?

Can anyone please help me with this, i got so far and do not want to give up. Any tips? anything.

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