==1.616121207

==1.617442799

==1.617851291

==1.617977531

==1.618016542

==1.618028597

According to the result, you can aware that a2 =

Then analyzing the formula, an+1 =

On the graph, it represents that at the point of the beginning it raises rapidly as acceleration. However, after that, an-an+1 value has been had no huge change which means that difference is close to 0.

Apply

Here is a proved formula

a=

a2=1+a

a2-a-1=0

Use quadratic equation

a= = 1.618033989 or －0.6180339887

However, the root cannot be the negative number

So,

∴ 1.618033989

Consider another infinite surd

where the first term is

Repeat the entire process above to find the exact value for this surd.

==1.847759065

==1.961570561

==1.990369453

==1.997590912

==.1999397637

==1.999849404

==1.999962351

==1.999990588

==1.999997647

==1.999999412

According to the result, you can aware that =

Then analyzing the formula, =

This graph shows us that there is a huge change between 1 and 2. And it is constant from 4. . bn and - is getting close to 0.

∴ bn- bn+1=0

Apply

Here is a proved formula

==

=2+

=--2=0

= (-2)*(

Again, the negative number is impossible in the root

Now consider the general infinite surd where the first term is. Find an expression for the exact value of this general infinite surd in terms of k.

So, the general formula would be =

Apply

Here is the solution

==

=

==k+

=--k=0

Use quadratic equation

The infinite surd value for k=0 will be 0. the infinite surd value can be 0 if k=0, but it can't be exactly 1.

If you take k=0,000... (m zeros)...0001000...(m zeros)...0001, the infinite surd value will be 1,000...(m zeros)...0001. So, you can obtain values as close to 1 as you want, but it will never be 1.

Find some values of k that make the expression an integer. Find the general statement that represents all the values of k for which the expression is an integer

The value of an infinite surd is not always an integer

2=1x2

6=2x3

12=3x4

20=4x5

30=5x6

These are proved, using this formula k= n (n+1)

5(5+1) = 30

5th term which is also described as 5x6=30

Test the validity of your general statement using other values of k.

First off, the formula is k= n (n+1)

Apply

With using quadratic equation,

8x9=72----→y =

=46

88x89=7832------→

=89

222x223=49506---------→

=223

And so on..

It is able to notice that

When k is 72, y = =46

Discuss the scope and/or limitations of your general statement.

According to this, we’ve found that value of K and n always are

n