- Level: International Baccalaureate
- Subject: Maths
- Word count: 1358
Math IA, Infinity surd
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Introduction
This is the infinity surd that is given on the example.
Finding a formula for in terms of :
First, since is the first term, let’s have a close look at the first few terms.
As you may can see, as the “n” gets larger, we can see a patter occurs. constantly added behind. We also can come up with a conclusion that, previous term is added behind at forward term. For example,
is equal to
From this conclusion, now we know the formula for , which is equal to
Calculating the values of the first tem terms of the sequence:
The values are given below
This graph shows the relationship between “n” and as “n” gets larger.
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
Middle
And put “lim” signs on both side,
And using the value that we have found above, when* i) -> “k”
ii) -> “k”
K = K+1
K -K -1 = 0
And we can solve this equation by using quadratic function.
K = 1 ± (-1)2–4(1)(-1)
2(1)
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
The first term is given, .
The exact pattern that we found above can be observed here, therefore the general formula is very similar.
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 as “n” gets larger
We can see from this graph, we have got the similar results as above. The same convergence is happening here as well.
Conclusion
The formula for Difference integer is,
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 as.
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 “” in this case)..
Therefore = 2.
And
Because the difference are the multiple numbers of 2
Therefore, = 2n.
Now let’s substitute the number’s in, 2 +
= 2 +
= 2 + (n-1)n
= n-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” = n -n +2
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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