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8below1994 · Jan 1, 2010, 11:48 PM

The sequence:
a1=√(1+√(1 )
a2=√(1+√(1+√(1 )
a3=√(1+√(1+√(1+√(1+... etc.

Calculate the decimal values of the first ten terms of the sequence. Using technology plot the relation between n and an. Describe what you notice. What does this suggest about the value of (a(n))-(a(n+1)) as n gets very large? Use your results to find the exact value for this infinite surd.

Unknown008 · Jan 2, 2010, 12:03 AM

If I got it right;

$a_1 = \sqrt(1+\sqrt1) \\ a_2 = \sqrt(1+\sqrt(1+\sqrt1)) \\ a_3 = \sqrt(1+\sqrt(1+ \sqrt(1+\sqrt1)))$

You'll have to grab a calculator for that, and find the value of each of the 10 terms.

For the second part, I can't help you. Apparently, you have a program which can plot the points of the sequence. Sorry.

For the third part, it's easy if you have plot the graph. You'll see that the value tends to be a constant and the graph makes an asymptote. I'll let you guess the answer to that.

Is it OK?:)

8below1994 · Jan 2, 2010, 12:07 AM

Makes sense, but what is the formula for a(n+1) in terms of a(n) ?

Unknown008 · Jan 2, 2010, 12:13 AM

Sorry, I don't see the question asking for the formula.

Think a little more now. If the terms approach a certain number, it means that they are nearly the same number, or roughly the same number if n is extremely large.

So, a_n is a term, where n is very large
a_(n+1) is the term just after a_n.

Can you see where I'm getting at?

8below1994 · Jan 2, 2010, 12:14 AM

Its making more sense... I asked a broader question... check that?

galactus · Jan 2, 2010, 05:09 AM

This is the nested radical form for the Golden Ratio.

Let L equal the limit. Then, since it is nested and convergent, we can write:

$L=\sqrt{1+L}$

$L^{2}-L-1=0$

$L=\frac{\sqrt{5}+1}{2}\approx 1.618$

If we let $L+1=\sqrt{1+(L+1)}$

We get $\frac{\sqrt{5}-1}{2}\approx 0.618$

Unknown008 · Jan 2, 2010, 11:19 AM

galactus, can you have a look at the other thread. It's in the same board and it's by the same member.

galactus · Jan 2, 2010, 12:08 PM

Let's do something general.

Start with $\sqrt{a+b\sqrt{a+b\sqrt{a+....}}}$

and assume it converges when $a\geq 0, \;\ b\geq 0$ .

$L=\sqrt{a+b\sqrt{a+b\sqrt{a+....}}}$

$L=\sqrt{a+bL}$

$L^{2}-bL-a=0$

Using the quadratic formula, we get

$L=\frac{b+\sqrt{b^{2}+4a}}{2}$

Now, for the given nested radical, a=b=1.

Therefore,

$L=\frac{1+\sqrt{1^{2}+4(1)}}{2}=\frac{\sqrt{5}+1}{ 2}$

The Golden Ratio, commonly designated as ${\phi}$