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# Infinite Surds

Extracts from this document...

Introduction

March 9, 08

Infinite Surds

1. (√1+√1+√1+√1+…)
 A1 √1+√1 1.4142 A2 √(1+ A1) 1.5537 A3 √(1+ A2) 1.598 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 ) An = √(1+ An -1)

An = √(1+ An )

An2  = 1 + An

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) 2 A10 √(2+ A9) 2
• As n increases, An also increases, however when n gets very large An - An  + 2
• An = √(2+ An -1)

Therefore:

• An+1 = √(2+ An )

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 :{ }

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