Difference of an and an+1
According to the chart, as the value of n increases, the difference between an and an+1 decreases. Base on this, we can predict that as the value of n becomes vey large, , and we can come up with an expression that represents as n approaches infinity.
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Equation for When
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Solving the Sequence
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To prove that the previous methods used are true for all infinite surds, we can apply them in a different situation.
In this case, the sequence would be expressed as ,, , etc.
Now we will try to find a formula for an+1 to show the relationship between an and an+1.
Formula for an+1 In Terms of an
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Decimal Values of the First Ten Terms of the Sequence
Below are the values of the first ten terms of the sequence accurate to the 9th place after the decimal point.
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Relation of an and an+1
Once again, as the value of n increases, the value of an seems to increase less. This is similar to what has happened with the previous example. The chart below describes the difference between an and an+1 as the value of n increases.
Difference of an and an+1
Compare to the previous example, the difference between an and an+1 is less in this case. However, the same pattern occurred. As the value of n increases, the difference between an and an+1 decreases. This means that when the value of n is very large . The expression , used to describe the difference between an and an+1 when for the last example, is also true for in this case, proving that is also a convergent sequence.
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Solving the Sequence
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General Infinite Surd
After looking at two examples of the infinite surds, we are ready to come up with some general statements with the general infinite surd below.
General Equation for an+1 in Terms of an
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Exact Value in Terms of k
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Restrictions of the expression
Before finding the values of k that will make the expression an integer, the restrictions has to be discussed. In this expression, the square root of the discriminate, , is an element of any odd positive integers, 0 or .
This restriction is stated due to the fact that an odd integer divided by 2 will result in a decimal instead of an integer. Since 1 add any even integer square root of will result in an old number, has to be an odd positive integer, “0”, or “”. The reason why that the discriminate has to be positive is because the square root of any negative number will result in an unreal solution. Examples are provided below to prove the restrictions.
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Some Values of k That Make the Expression an Integer
By allowing the expression to equal to some positive integers, and then solve for “k”, we are able to find some values of “k” that makes the expression an integer.
A general pattern can be found from the chart that represents relation between the value of “k” and the value of the expression.
Relation between Value of “k” and Value of the Expression
The formula represents the relations between “k” and the value of the expression. To prove this, we can substitute 6 as “x”, the value of the expression, which will give us, , and . This result is the same as the value in the chart. We can also try to put the number 9 into x’s place, resulting in, which will give us . This is also the same as the result in the chart. We can factor the formula and this will result in . By looking at the formula, we can tell that x multiply a number that is one lower than itself will equal to k. This means that the product of two consecutive numbers equals to k. The restriction for this is that x has to be an integer. This is because only the product of two integers will be an integer.
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Proving the General Statement
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How Big Can and Be?
Right now, we know that for the expression to be an integer, “k” has to be the product of any two consecutive integers, and we also know that . Now, we will consider a special case for k. When, the value of the expression should also be infinity. This is shown by the calculations below.
We know that the calculations are true because . So this means that both the value of k and can be infinity. So the following conclusion can be made. , and
Conclusion
By first taking a look at two simple infinite surds, we are able to see some similarities in them. Then we are able to apply what we learnt into coming up with the general statements of infinite surds. Graphs and charts were used to aid and made the understanding of infinite surds easier. In conclusion the following general statements about surds can be made:
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The relation between an and an+1 for all sequences of infinite surds can be represented by .
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The expression can be used to represent the exact value of all infinite surds. The restrictions are, , and that can only be a positive odd integer, 0 or .
- Infinite surds all have a convergence.
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The formula can be used to calculate the values of “k” that make the infinite surd an integer. The product of any two consecutive integers will allow the infinite surd to be an integer.
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In general, for the infinite surd to have a real solution, , and the value of the infinite surd when k is 0,, or a positive integer is .
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