Alice Wang

Infinite Surds

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An infinite surd is a number that can only be expressed exactly using a root sign. Surds are useful because they can represent irrational numbers that have an infinite number of non-recurring decimals.

This expression is known as an infinite surd:

It can be considered as a sequence of terms an, where


It can be inferred from the first ten terms of the sequence that the relation of two consecutive terms is .

Figure 1. Relationship between n and an

From Figure 1, it can be inferred that the sequence of the two following terms as:

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. The values of an, are not exact because the values of an, have an infinite number of decimal points since it is a surd. When the value of n increases, the difference between two consecutive values of n are smaller than

Graph 1

Graph 1 exemplifies the direct relationship between an and n. As the value of n increases, the value of an increases. In other words, as n gets larger, the value of  gets smaller. When n gets larger and gets closer to infinity, an increases at a slower rate (ex:).  However, the difference between an and n will ...

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