Sequences

[intervade]

Ok, my problem is as follows:

Given the sequence(which is convergent) defined recursively by

\(\displaystyle a_{1}=\sqrt{5}\)
\(\displaystyle a_{n+1}=\sqrt{5+a_{n}}\)

Now, it says list the first 4 terms of the sequence.. does this mean..

\(\displaystyle a_{2}=\sqrt{5+\sqrt_{5}}\)
\(\displaystyle a_{3}=\sqrt{5+\sqrt_{5 + \sqrt_{5}}}}\)

etc..

Also, I am suppose to find if the limit of the sequence. I have not done this with a recursively defined sequence before, so I'm not sure how to go about doing that.

Help on this would be much appreciated!

 

[stapel]

Given the sequence(which is convergent) defined recursively by

\(\displaystyle a_{1}=\sqrt{5}\)
\(\displaystyle a_{n+1}=\sqrt{5+a_{n}}\)

Now, it says list the first 4 terms of the sequence.. does this mean..

\(\displaystyle a_{2}=\sqrt{5+\sqrt_{5}}\)
\(\displaystyle a_{3}=\sqrt{5+\sqrt_{5 + \sqrt_{5}}}}\)

That is certainly how I would read this.

I am suppose to find if the limit of the sequence.

The limit will be the "infinitely-many" form:

. . . . .\(\displaystyle \sqrt{5\, +\, \sqrt{5\, +\, \sqrt{5\, +\,...}}}\)

...with the "dot, dot, dot" meaning "forever and ever onwards".

Once you're dealing with "forever", you can use some tricks, because deleting one item from the "front end" of "forever" doesn't change the "forever-ness" of what is left. So you name the "forever" value as being:

. . . . .\(\displaystyle x\, =\, \sqrt{5\, +\, \sqrt{5\, +\, \sqrt{5\, +\,...}}}\)

Now the trick for using this name is to note the following:

. . . . .\(\displaystyle \sqrt{5\, +\, \sqrt{5\, +\, \sqrt{5\, +\,...}}}\, =\, \sqrt{5\, +\, \left(\sqrt{5\, +\, \sqrt{5\, +\,...\right)}}}\, =\, \sqrt{5\, +\, x}\)

By definition of \(\displaystyle x\), we then have:

. . . . .\(\displaystyle x\, =\, \sqrt{5\, +\, x}\)

Square both sides, and solve for the value of \(\displaystyle x.\) :wink:

 

[intervade]

Ok, that makes perfect sense except for the..

\(\displaystyle x=\sqrt{5+x}\)

Are you saying that these value(s) of x are what this sequence converges to?

 

[stapel]

Ok, that makes perfect sense except for the..

\(\displaystyle x=\sqrt{5+x}\)

Are you saying that these value(s) of x are what this sequence converges to?

Isn't that how \(\displaystyle x\) was defined?

Does this help?

. . . . .\(\displaystyle x\, =\, \sqrt{5\, +\, \sqrt{5\, +\, \sqrt{5\, +\,...}}}\, =\, \sqrt{5\, +\, \left(\sqrt{5\, +\, \sqrt{5\, +\,...\right)}}}\, =\, \sqrt{5\, +\, x}\)

Then take out the middle steps to get:

. . . . .\(\displaystyle x\, =\, \sqrt{5\, +\, x}\)