MAth portfolio-Infinite surds

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Samonvye Reddy| {Math Portfolio}                    

Q1) Find a formula for an+1 in terms of an.

Answer)

The following expression is an example of an infinite surd:

This means that can me shown as=

a1 =

a2 = 

a3 = 

A formula for an+1 in terms of an would be:

a2 =

a3 =

Square on both sides

(a3)2 = ()2

(a3)2 = 1 + a2

a2 = (a3)2 – 1

an = (an+1)2 – 1

(an+1)2 = an + 1

Therefore,

an+1 =

Q2) Calculate the decimal values of the first ten terms of the sequence. Using technology, plot the relation between ‘n’ and ‘a’. Describe what you notice. What does this suggest about the value of   an – an+1 as ‘n’ gets very large? Use your results to find the exact value for this infinite surd.

Ans)

  • n                        Term (an)                          an – an+1                
  • 1                        1.414213562                                  -
  • 2                        1.553773974                        0.139560412
  • 3                        1.598053182                        0.044279208
  • 4                        1.611847754                        0.013794572
  • 5                        1.616121207                        0.004273452
  • 6                        1.617442799                        0.001321592
  • 7                        1.617851291                        0.000408492
  • 8                        1.617977531                        0.000126241
  • 9                        1.618016542                        3.90113 x 10-5
  • 10                        1.618028597                        1.20552 x 10-5
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The difference between each successive term is beginning to decrease and is almost at zero.

 As ‘n’ keeps increasing, the value of an – an+1 keeps decreasing , and gradually at that.

The exact value of this infinite surd is 1.62.

Q3) Consider another infinite surd,  where the first term is  Repeat the entire process above and find the exact value for this surd.

Answer) Basically repeat the same process for the first infinite surd that had 1, instead of 2.

So, in this case we take the surd as a sequence of terms, where   is;

a1 ...

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