Infinite Surds Coursework

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Petri Alexia

12T1

Standard Maths

Infinite Surds Coursework

The following expression is an example of an infinite surd.

Find the formula for an+1 in terms of a 

a1 =

a2 =         a2 =

a3 =         a3 =

an+1 =

an =

Calculate the decimal values of the first ten terms of the sequence

a1 = 1.414213562373100

a2 = 1.553773974030040

a3 = 1.598053182478620

a4 = 1.611847754125250

a5 = 1.616121206508120

a6 = 1.617442798527390

a7 = 1.617851290609670

a8 = 1.617977530934740

a9 = 1.618016542231490

a10 = 1.618028597470230


Using technology, plot the relation between n and an. Describe what you notice.

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By plotting the relation between n and an, one notices that as n increases, an increases. However this increase is not proportional to the increase of n, an seems to be increasing towards 1.62.

Once n reaches 28 an ceases to increase, remaining stable at 1.618033988749890.

This suggests that as n becomes very large an – an+1 = 0

As such, we can conclude that the exact value for this infinite surd is 1.618033988749890.

Consider another infinite surd:

 

Find the formula for an+1 in terms of a 

a1 =

a2 =         a2 =

a3 =         a3 =

an+1 =

an = ...

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