Introduction
In this project I will investigate the infinite surds. A surd is a number that cannot be changed into a fraction. They go on infinitely without any pattern. They are usually a square root of a number.
My project will include two formulas. One is the formula of an+1 in terms of an. The second formula will represent the general statement that represents all the values of K for which the expression is an integer.
Terms, scopes and limitations
The formula of an+1 in terms of an
The formula for an+1 in terms of an =
U1= √(1+√1)
= √2
U2= √(1+(√1+√1))
= √(1+√2)
U3= √(1+(√1+√1))
= √(1+√(1+√2))
Un=√(1+Un-1)
Un+1=√(1+Un)=
Un+1=√(K+Un)
Un =1
A1=√(1+√1)
= 1.414213562
A2=√(1+(√1+√1))
= 1,553773974
A3=√(1+(√1+(√1+√1)))
= 1,598053182
A4=√(1+(√1+(√1+(√1+√1))))
= 1,611847754
A5=√(1+(√1+(√1+(√1+(√1+√1)))))
= 1,616121207
A6=√(1+(√1+(√1+(√1+(√1+(√1+√1))))))
= 1,617442799
A7=√(1+(√1+(√1+(√1+(√1+(√1+(√1+√1)))))))
= 1,617851291
A8=√(1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+√1))))))))
= 1,617977531
A9=√(1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+√1)))))))))
= 1,618016542
A10=√(1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+(√1+√1))))))))))
= 1,618028597
The relation between n and an is shown if an – a1+2. For example U2= √(1+√2 – U3)
= √(1+√(1+√2))
= (√1+√2) – (√1+√1+√2) = U1
(an = (√1+√2) –
a1+2 = (√1+√1+√2))
= √2.
( = an-1)
an – a1+2 = an-1
so Un=√2+Un-1 and Un+1
=√2+Un
For example
U2= √(2+(√2 - U3))= √(2+(√2+√2))
= (√2+√2) – (√(2+(√2+√2))) = U1
= √2.
an – a1+2 = an-1
Un =2
A1=√(2+√2)
= 1,414213562
A2=√(2+(√2+√2))
= 1,847759065
A3=√(2+(√2+(√2+√2)))
= 1,961570561
A4=√(2+(√2+(√2+(√2+√2))))
= 1,990369453
A5=√(2+(√2+(√2+(√2+(√2+√2)))))
=1,997590912
A6=√(2+(√2+(√2+(√2+(√2+(√2+√2))))))
= 1,999397637
A7=√(2+(√2+(√2+(√2+(√2+(√2+(√2+√2)))))))
= 1,999849404
A8=√(2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+√2))))))))
=1,999962351
A9=√(2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+√2))))))))
=1,999990588
A10=√(2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+√2)))))))))
= 1,999997647
Un =3
A1=√(3+√3)
= 1,732050808
A2=√(3+(√3+√3))
= 2,175327747
A3=√(3+(√3+(√3+√3)))
= 2,274934669
A4=√(3+(√3+(√3+(√3+√3))))
= 2,296722593
A5=√(3+(√3+(√3+(√3+(√3+√3)))))
= 2,301460969
A6=√(3+(√3+(√3+(√3+(√3+(√3+√3))))))
= 2,302490167
A7=√(3+(√3+(√3+(√3+(√3+(√3+(√3+√3)))))))
= 2,302713653
A8=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3))))))))
= 2,302762179
A9=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3)))))))))
=2,302772715
A10=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3))))))))))
=2,302775003
Un =4
A1=√(4+√4)
= 2,000000000
A2=√(4+(√4+√4))
= 2,449489743
A3=√(4+(√4+(√4+√4)))
= 2,539584561
A4=√(4+(√4+(√4+(√4+√4))))
= 2,557261144
A5=√(4+(√4+(√4+(√4+(√4+√4)))))
= 2,560714967
A6=√(4+(√4+(√4+(√4+(√4+(√4+√4))))))
= 2,561389265
A7=√(4+(√4+(√4+(√4+(√4+(√4+(√4+√4)))))))
= 2,561520889
A8=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4))))))))
= 2,561546581
A9=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4)))))))))
=2,561551596
A10=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4))))))))))
=2,561552575
K=5
A1=√(5+√5)
= 2,236067977
A2=√(5+(√5+√5))
= 2,689994048
A3=√(5+(√5+(√5+√5)))
= 2,773083852
A4=√(5+(√5+(√5+(√5+√5))))
= 2,788025081
A5=√(5+(√5+(√5+(√5+)√5+√5)))))
= 2,790703331
A6=√(5+(√5+(√5+(√5+(√5+(√5+√5))))))
= 2,791183142
A7=√(5+(√5+(√5+(√5+(√5+(√5+(√5+√5)))))))
= 2,791269092
A8=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5))))))))
= 2,791284488
A9=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5)))))))))
=2,791287246
A10=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5))))))))))
=2,791287740
Un =100
A1=√(100+√100)
= 10,000000000
A2=√(100+(√100+√100))
= 10,488088482
A3=√(100+(√100+(√100+√100)))
= 10,511331432
A4=√(100+(√100+(√100+(√100+√100))))
= 10,512436988
A5=√(100+(√100+(√100+(√100+(√100+√100)))))
= 10,512489571
A6=√(100+(√100+(√100+(√100+(√100+(√100+√100))))))
= 10,512492072
A7=√(100+(√100+(√100+(√100+(√100+(√100+(√100+√100)))))))
= 10,512492191
A8=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+√100))))))))
= 10,512492197
A9=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+√100)))))))))
=10,512492197
A10=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100))))))))))
=10,512492197
Uk= √(k+(√k – Uk+2))= √(k+(√k+√k))
= (√(k+√k)) – (√(k+(√k+√k))) = Uk-1
= √2.
Ak – ak+2 = ak-1
The general statement that represents all the values of K for which
The expression is an integer
U1=√(k+√k)
U2=√(k+(√k+√k))
U3=√(k+(√k+(√k+√k)))
U1000000=√(k+U999999)
Un+1=√(k+Un)
√(k+)
+1
2K+
2-=
X2- X-K=
1+√1)=√2
e.g.
…4
=2
Working steps
(K+K)=4
=2
K = 4
K=(4K)2
=(4K) X (4K)
=16K4K+K2
=168K+K2
Given as a formula 16 = K
8K = X
K2 = X2
X2XK=0
Plot the formula in the graphic calculator in math solver enter green alpha enter
=12.468871125
K=12.468871125
=12.468871125 +(12.468871125)) =4
Excel evidence
Conclusion
The formula of an+1 in terms of an =
Un+1=√(K+Un) Which is explained in the first chapter.
The general statement that represents all the values of K for which the expression is an integer =
X2XK=0 Which is explained in the second chapter.
Bibliography
