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Infinite Surds

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Introduction

March 9, 08

Infinite Surds

  1. (√1+√1+√1+√1+…)

A1

√1+√1

1.4142

A2

√(1+ A1)

1.5537

A3

√(1+ A2)

1.5980

A4

√(1+ A3)

1.6118

A5

√(1+ A4)

1.6161

A6

√(1+ A5)

1.6174

A7

√(1+ A6)

1.6178

A8

√(1+ A7)

1.6179

A9

√(1+ A8)

1.61801

A10

√(1+ A9)

1.61802

  • As n increases, An also increases, however when n gets very large An - An  + 1
  • An = √(1+ An -1)
  • An+1 = √(1+ An )                        

image00.png

An = √(1+ An -1)                        

An = √(1+ An )

An2  = 1 + An

Quadratic formula :

...read more.

Middle

1 +√(5) / 21 -√(5) / 2 < 0answer: 1 +√(5) / 2

An2  - An - 1 = 0  : As n gets very large An - An-1 = 0 ; An = An-1

2.) √(2+√2+√2+√2+…)

A1

√2+√2

1.8477

A2

√(2+ A1)

1.9615

A3

√(2+ A2)

1.9903

A4

√(2+ A3)

1.9975

A5

√(2+ A4)

1.9993

A6

√(2+ A5)

1.9998

A7

√(2+ A6)

1.99996

A8

√(2+ A7)

1.99999

A9

√(2+ A8)

1.999997

A10

√(2+ A9)

1.999999

  • As n increases, An also increases, however when n gets very large An - An  + 2
  • An = √(2+ An -1)

Therefore:

  • An+1 = √(2+ An )                        
...read more.

Conclusion

n increases by jumping even numbers, it equals to an integer.  Sequence value of k:
  • k = 2
  • k = 6
  • k = 12
  • k = 20
  • k = 30

The difference between each value is 2, 4, 6, 8, 10

  • 1 + 4k needs to be a square number and it has to be odd.

So therefore if k is 2 in 1+4k then:

  • When k = 2

1+4k = 1 + 4(2) = 9

  • When k = 6

1+4k = 1+4(6) = 25

  • 4, 9, 16, 25 , 36…
  • k can not be below 0 and you can not square root a negative number so therefore.
  • k :{ image02.png}

...read more.

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