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Math IA, Infinity surd

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Introduction

image00.png

This is the infinity surd that is given on the example.

Finding a formula for image01.pngin terms of image12.png:

First, since image17.pngis the first term, let’s have a close look at the first few terms.

image23.png

As you may can see, as the “n” gets larger, we can see a patter occurs. image32.pngconstantly added behind. We also can come up with a conclusion that, previous term is added behind at forward term. For example,

image36.pngis equal to image44.png

From this conclusion, now we know the formula for image01.png, which is equal to image18.png

Calculating the values of the first tem terms of the sequence:

The values are given below

image02.png

image03.png

image04.png

image05.png

image06.png

image07.png

image08.png

image09.png

image10.png

image11.png

This graph shows the relationship between “n” and image12.png as “n” gets larger.

image13.png

From the results we have from values of first 10 terms, and the graphs, we can see that there is a convergence occurs. The value from 1 to 3 increases rapidly, but after that, the number increases in a small numbers. From 4th dot, from just by looking at the graph, we hardly can see any changes after that. But we still can see a small number is increasing by looking at the values I have worked out above. As this is an infinity surd, we know that this image12.png

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Middle

image19.png.

And put “lim” signs on both side, image20.png

And using the value that we have found above, when* i) image15.pngimage12.png-> “k”        

                                                        ii) image15.pngimage16.png-> “k”

image21.pngKimage22.png = K+1

                                                   Kimage22.png -K -1 = 0

And we can solve this equation by using quadratic function. image24.jpg

K =   1 ±    (-1)2–4(1)(-1)

                2(1)

image24.jpg

K =  1 ±     5

            2

K ≈ 1.618033989

Now, there is a positive value and negative value, which one is right..? the positive one will be the most logical answer. Because all of the values in the graph above are above zero, And the values in square roots were positive. Obviously, “K” value is positive.

We now have worked out the exact value(In other words, the Limit), therefore we know that the number will be keep increasing so close to the number, but it will never reach it.

Consider another infinity surd image25.png

The first term is given, image26.png.

image27.png

The exact pattern that we found above can be observed here, therefore the general formula is very similar.

image28.png

Use the same method, the first 10 series of values and the graph is:

b1 ≈ 1.8477590650226

b2 ≈ 1.9615705608065

b3 ≈ 1.9903694533444

b4 ≈ 1.9975909124103

b5  ≈ 1.9993976373924

b6  ≈ 1.9998494036783

b7  ≈ 1.9999623505652

b8  ≈ 1.9999905876192

b9  ≈ 1.9999976469034

b10≈ 1.9999994117258

Relationship between “n” and image29.png as “n” gets larger

image30.png

We can see from this graph, we have got the similar results as above. The same convergence is happening here as well.

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Conclusion

image35.pngandimage37.pngis 4, and difference between image38.png and image35.png is 6, and the difference between image39.png and image38.pngis 8, and so on. See the numbers of difference, 4,6,8, …… the numbers are constantly increasing by 2. This is telling us we can use “Difference integer” formula to find the general formula that represents all the values of “k” for which the expression is an integer.

The formula for Difference integer is,

image40.png

The “n” has to be always greater or equal to 2 because “n” cannot be equal to zero, due to the fact that there is no such thing asimage41.png.

Okay, we have 2,6,12,20,30, … … value for “k”,(the numbers which will always make the expression an integer)

And 4,6,8,10, … … for Difference(which is “image42.png” in this case)..

Therefore image43.png= 2.

And

Because the difference are the multiple numbers of 2

Therefore,  image42.png= 2n.

Now let’s substitute the number’s in, 2 + image45.png

                                   = 2 + image46.png

                                   = 2 + (n-1)n

                                   = nimage22.png-n +2

Here we go, now we have worked out the general statement that represents all the values of “k” for which the expression is an integer.

Final conclusion is,      “k” = nimage22.png -n +2

...read more.

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