If it's a Fibonacci like sequence, then that means that each term (starting with the 3rd) is sum of previous two terms. So we have:
x, 8, x+8, x+16, 2x+24, 3x+40, ...
5th term = 27
5th term = 2x+24
2x + 24 = 27
2x = 3
x = 1.5
So first 6 terms are:
1.5, 8, 9.5, 17.5, 27, 44.5, ...

Here are the first twenty terms (and ratios) for the sequence:
n Fn Ratio
1 1
2 1 1
3 2 2
4 3 1.5
5 5 1.666666667
6 8 1.6
7 13 1.625
8 21 1.615384615
9 34 1.619047619
10 55 1.617647059
11 89 1.618181818
12 144 1.617977528
13 233 1.618055556
14 377 1.618025751
15 610 1.618037135
16 987 1.618032787
17 1597 1.618034448
18 2584 1.618033813
19 4181 1.618034056
20 6765 1.618033963
The ratio is approaching (1 + √5)/2, which is the Golden ratio. A rectangle is said to have the most pleasing proportions if its height and width have this ratio.
(1 + √5)/2 = 1.618033989

Zen is somewhat based on this. As any ratio indicates there is a balance in life. If you examine the food pyramid you'll see that the Golden ratio applies. It shows up in many of our activities. You can follow it by basing everything on it. Activities, time, locations, all can be made to conform to the ratio.
There is another "golden ratio" called Parado's Principle. I have based much of my life on it and it generally holds true. The principle is simple -80/20. 80% is governed by 20%. For instance: 80% of the money is held by 20% of the people.

Ratio of two consecutive terms of the Fibonacci sequence is always rational (ratio of two integers), but the golden ratio [(1+√5)/2] is an irrational number. So they can never be equal.
[But the ratio of two consecutive terms of the Fibonacci sequence tends to the golden ratio when n tends to infinity.]

You could have the sequence consisting of:
1, 2, 2, 4, 8, 32, 256, 8192...
that is the sequence defined by
x_1 = 1
x_2 = 2
x_n = x_{n-1} * x_{n-2}
or the Fibonacci sequence but with multiplication. This may be too easy since it is really just the same. Maybe this would be good for the test just as a "were you paying attention" check.
There are *tons* of sequences over on the "On-Line Encyclopedia of Integer Sequences" but you'd have to pick some that have a nice computational property.
One that you can use to talk about numbers that grow very large very quickly is the Ackermann function. This function is pretty easy to code, it is just two recursive calls, like Fibonacci, but it gets *crazy huge* even for fairly small input numbers. In fact, this function is the standard and semi-snotty retort whenever anyone argues about simple functions not being complex, or someone has a new compiler that you want to try and crush.
Another would be to find perfect numbers. These are numbers where when you take half the sum of its proper divisors, you get the number itself. Examples are: 6 (1+2+3+6)/2, 28 (1+2+4+7+14+28)/2... You could even mention that it is an open question in mathematics whether or not there are any odd perfect numbers.
Lastly, an interesting bunch of numbers are "friendly numbers." Two numbers are friendly if they share the same ratio of their divisors to the number itself:
6 and 28 are friendly because:
(1+2+3+6)/6 = 2
and
(1+2+4+7+14+28)/28 = 2
You can even reuse some of your code that you wrote for finding divisors of a number! Code reuse, what a great lesson.

See this logic in Matlab.
It's an almost direct-conversion to VB
http://matrixlab-examples.com/fibonacci-numbers.html
.

Leonardo of Pisa known as Filius Bonacci (son of Bonacci). His father was an Italian merchant who traded in North Africa which is where the boy grew up and became exposed to Arabic mathematics. It was through contacts like this that Arabic and Indian mathematical ideas entered mainstream European culture,
He was led to the sequence by considering the problem of the growth of a rabbit population and published his findings in his influential book Liber Abaci (Book of the Abacus) in 1202,
He considered the growth of an idealised (biologically unrealistic) rabbit population, assuming that:
(a) in the first month there is just one newly-born pair,
(b) new-born pairs become fertile from their second month on
(c) each month every fertile pair begets a new pair, and
(d) the rabbits never die
He argued thus: Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2).
(3) And that is the generating rule for the Fibonacci sequence. Each term is the sum of the previous two terms.
(2) Conventionally it begins with 0, 1 and continues 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 ...
(4) The Golden Ratio, Phi 1.618033989 .., and its inverse phi (0.618033989 ...) are both produced by considering how the ratios of successive terms of the Fibonacci sequence converge (from either side) on the Golden Ratio.
3/5 = 0.6
5/8 = 0.625
8/13 = 0.61538 ...
13/21 = 0.61904 ...
21/34 = 0.61765 ...
34/55 = 0.61818 ...
55/89 = 0.61797 ...
89/144 = 0.61805 ...
10946/17711 = 0.618033990 ...
17711/28657 = 0.618033988 ...
AND SIMILARLY
5/3 = 1.666 ...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538 ...
17711/10946 = 1.618033985 ...
28657/17711 = 1.618033990 ...
(5) The two best-known examples of Fibonacci numbers appearing in nature are in the arrangements of sunflower heads and pine cones.

I used a TI-86, but I'm guessing you would like to use a computer code to do this.
For what it's worth, the TI-86 produces the result: 4.07466709145E412
using the algorithm: F=((((1+sqrt(5))/2)^1976)/sqrt(5))+.5
EDIT: whipping up a quick matlab script did not turn out to be fruitful but may get you started. This code reached an integer limit in Matlab I believe.
EDIT2: OK, one more try.
This code will produce the results you want with Matlab and the Symbolic Toolbox using the command 'vpa' (taken from link below)
>> A=[0,1;1,1]
>> vA = vpa(A,30)
>> vA^10000
These are usefull resources: (check the link to the matlab group)
http://www.congruentialuminaire.com/cLBlog/Lists/Posts/Post.aspx?ID=3
http://www.congruentialuminaire.com/cLBlog/Lists/Posts/Post.aspx?ID=5

The first two terms in the Fibonacci sequence are:
1, 1
Every subsequent term is formed by adding the two before it:
1, 1, 2
1, 1, 2, 3
1, 1, 2, 3, 5
1, 1, 2, 3, 5, 8
etc.
This is similar to Pascal's triangle, where the outside is formed of 1s, and each internal cell is formed by adding the two above it.

Okay, φ is defined to be the solution to: x^2 - x - 1 = 0
Hence φ^2 = φ + 1
So φ^3 = φ(φ + 1) = φ^2 + φ = 2φ + 1
φ^4 = 2φ^2 + φ = 2(φ + 1) + φ = 3φ + 2
...
Do you see where I'm going with this?
It is clear that if we continue multiplying by φ and simplifying, we can always write φ^n as Aφ + B
Well, what exactly are A and B?
Well, since we have an inductive sort of method, it makes sense to work out what φ^n is in terms of smaller powers:
φ^n = φ^(n-2) * φ^2 = (φ+1)φ^(n-2) = φ^(n-1) + φ^(n-2)
So the nth power is the sum of the previous two powers. Does this surprise you? It is exactly how the Fibonacci sequence works! Therefore you should be able to see that A and B are in fact Fibonacci numbers in the sequence.
Well, for n = 1, A = 0 and B = 1
for n = 2, A = 1 and B = 1
and using these initial conditions you can see that the A's will follow the sequence: 0, 1, 1, 2, 3, 5, ...
And the B's will go: 1, 1, 2, 3, 5, ...

There is no intrinsic relationship as far as I know, and I did a paper on it in college. The Fibonacci Series Ratio is the music ratio, determining the relationships between musical instruments in the orchestra, although it has many other applications.
Look up Isaac Asimov's book Science, Numbers, and I for a more detailed examination of the Fibonacci Ratio--Bry

http://en.wikipedia.org/wiki/Fibonacci_number

The 6th term of the standard Fibonacci sequence = 5.
Binet's formula is F_n = (1/sqrt5) (phi^n - psi^n). Hence for n = 5
F_5 =(1/sqrt5) [ ((1/2(1 + sqrt5)^5 - (1/2(1-sqrt5)^5)]
F-5 = 5 as was to be shown
Is that sufficient? Just be careful with the arithmetic. It's all a bit too long to type here but get back if you get stuck or can't get it to work out.
Oh, just spotted what you said that the 6th term is 8. I make it the 7th term:
The Fibonacci numbers are:
0,1,1,2,3,5,8,13, ...

uh.. i think this is what you want..
Dim seq As New System.Collections.Generic.List(Of Integer)
seq.Add(0)
seq.Add(1)
Dim going As Boolean = True
While seq(seq.Count - 1) < 2000
seq.Add(seq(seq.Count - 1) + seq(seq.Count - 2))
End While
Dim finalnumber As Integer = 0
For Each num As Integer In seq
If (num Mod 2 = 0) Then
finalnumber += num
End If
Next
MsgBox(finalnumber)
Assuming i understand the question.. finalnumber = 3382
the program adds 0,1 to the list.. and loops through 'till the last number in the sequence is >= 2000 while adding the two previous in the list.
then in loops through the sequence and adds up the even numbers.

I don't know, but there is a simple closed form for the Fibonacci numbers and the number of bitwise operations required does not grow very quickly so there would be few impediments to its calculation.
http://upload.wikimedia.org/math/9/6/8/968be88f42e32712cb10d89a765ce708.png

Plants form spiral to achieve efficient packaging or surface exposure to sunlight. The arrangement of seed in the composite head of a sunflower follow radiating spirals. Cones package seeds in a spiral of scales while stems spiral branches and leaves to avoid self-shading. Some 90% of plants show a Fibonacci phyllotaxis (leaf arrangement moving out along the stem) with the number of elements positioned being successive elements in the Fibonacci sequence.
The spiral of leaves around a stem can follow a Fibonacci sequence of spacing as the stem elongates. Apical meristem cells, which will later develop into organs like leaves or flower petals usually form in the least crowded spot along the growing tip of a plant. The plant grows placing the second cell as far as possible from the first, and the third is placed at a distance farthest from both the first and the second cell in the bud. As the number of cells increases, the divergence angle between each successive cell eventually converges to a constant value of 137.5 degrees and thus creates Fibonacci spirals.
The spirals are not a law of nature but are a fascinatingly prevalent tendency of cellular growth in a limited space.
Fibonacci in nature
http://goldennumber.net/nature.htm
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#plants

There are infinitely many terms. Each term is the sum of the two previous terms, so there never is any last one.

It is called Binet's formula. You can find it here:
http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html
Any sequence whose n-th term is computed based on the previous two terms might be considered 'similar' to the Fibonacci sequence. For example:
1, 2, 4, 8, 16, 32, 64, ...
Here the n-th term is 2 times the (n-2)-th term, plus the (n-1)-th term.
Another example:
0, 3, 3, 6, 9, 15, 24, 39, ...
Here, like the Fibonacci sequence, each term is the sum of its two closest predecessors.

The Fibonacci sequence begins with 1, and the second term is 1, and each term after that is the sum of the two before it.
So the first five terms of the FIbonacci sequence would be 1, 1, 2, 3, 5, for example.
It is possible to write this recursively, but recursion doesn't use a while (Recursion takes a problem and breaks it up into an "exit condition" (like "if n == 1 return 1; else if n == 2 return 1)" e.g. conditions that make it terminate, and also one or more larger conditions that break the problem down into smaller pieces that are guaranteed to eventually lead to an exit condition, like "... else return fibonacci( n-1 ) + fibonacci( n-2 )" )
To use a while loop saves you the runtime and stack frames that recursion would cost you. This would be the "iterative" version (which is what uses a while). Note that to do this, you have to have it "remember" the previous two terms.
The approach to write the iterative version is similar. You want to think about what will make your loop exit, and how to start off. Since the assignment tells you the maximum terms permitted should be between 1 and 50 (I assume inclusive), I interpret that to mean they want you to check for that, yourself.
So you could begin with something like "if( n >= 1 && n <= 50 ) { /* entire fibo routine here */ }"
public class CheckFibo {
public static void fibo ( int n ) {
if( n >= 1 && n <= 50 ) {
int prev1 = 1; int prev2 = 1; // previous term and one before that
int i = 3; // counts up terms; "i" will start at the 3rd term
if( n > 0 ) // print out first term, as long as at least one term requested
System.out.print( prev2 + " " );
if( n > 1 ) // print out second term, if requested
System.out.print( prev1 + " " );
while( i <= n ) { // compute and print remaining terms, if any
int curr = prev1 + prev2; // calculate current term
System.out.print( curr + " " ); // output it
prev2 = prev1; // prev term now becomes penultimate term
prev1 = curr; // term just computed now becomes "prev" term
i++; // ith term just printed, so increment i
}
}
}
public static void main( String[] args ) {
int n = 7; // print n terms of Fibonacci series
System.out.println( "printing first " + n + " terms of Fibonacci series" );
fibo( n );
}
}

____________________
F(1)=1,
F(2)=1,
F(n+2)=F(n+1)+F(n), n≧１
_____________________
x^2=x+1, x^2-x-1=0
A=[(1+√5)/2], B=[(1-√5)/2]
A+B=1
AB=-1
F(n+2)-A・F(n+1)=B[F(n+1)-A・F(n)]
F(n+2)-B・F(n+1)=A[F(n+1)-B・F(n)]
because,
F(n+2)-(A+B)F(n+1)+AB=0
F(n+2)-F(n+1)-F(n)=0
F(n+2)=F(n+1)+F(n)
__________________________________________
F(n+1)-A・F(n)=｛F(2)-A・F(1)｝・B^(n-1)
F(n+1)-B・F(n)=｛F(2)-B・F(1)｝・A^(n-1)
F(2)-A・F(1)
=1-[(1+√5)/2]=[(1-√5)/2]=B
F(2)-B・F(1)
=1-[(1-√5)/2]=[(1+√5)/2]=A
F(n+1)-A・F(n)=B^n
F(n+1)-B・F(n)=A^n
(A-B)・F(n)=A^n - B^n
A-B=[(1+√5)/2]-[(1-√5)/2]=√5
F(n)=[A^n-B^n]/(A-B)
in this case , the solution :
F(n)=[｛(1+√5)/2｝^n - ｛(1-√5)/2｝^n]/√5
_______________________
a bit of verification ,
F(1)
=[｛(1+√5)/2｝- ｛(1-√5)/2｝]/√5
=√5/√5=1
F(2)
=[｛(1+√5)/2｝^2 - ｛(1-√5)/2｝^2]/√5
=[(6+2√5)-(6-2√5)]/(4√5)
=(4√5)/(4√5)=1
F(3)
=[｛(1+√5)/2｝^3- ｛(1-√5)/2｝^3]/√5
=[(1+3√5+15+5√5)-(1-3√5+15-5√5)]/(8√5)
=(16√5)/(8√5)=2 .
______________

Have you considered an investigation into whether sequences similar to the main Fibonacci one behave in the same way.
Try sequences that have the same rule, i.e. each term the sum of the previous two, but that don't start with 1, 1, e.g.
2, 7, 9, 16, 25, 41, 76, . . .
-5, -1, -6, -7, -13, -20, . . .
Does each term divided by the previous one work towards a limit like the golden number for the main Fibonacci sequence? You might be surprised by the answer.
You can go even more weird. What about a sequence where each term is the sum of the one two before and the one three before. You will need to start with three terms. In the one below I made up 2, 5, 6 to start with.
2, 5, 6, 7, 11, 13, 18, 24, 31, 42, . . .
Then there are the practical applications of Fibonacci sequences such as the prongs on a pine-cone. Type "Fibonacci sequence" into google and you will get far more than you need.
Yes, it is a fascinating topic.

Try WIki
http://en.wikipedia.org/wiki/Fibonacci_number

The area sums is based on starting with 1, then take the longer side (1) and make a square which will be 1 + 1 = 2, and the long side = 2, add a square 2 x 2 to the rectangle 1 x 1, the resulting longer side = 3, so add a 3 x 3 square, and the longer resulting side would be 5, so add a square 5 x 5 and so on, and so on......

Never exactly. The very simple Fibonacci sequence has a repetitive fractal like growth, meaning simply that each step builds on a reflection of the proportions of the previous steps without the need for additional information. This is similar to some natural growths -- particularly growths that follow the path of least resistance or evolved to have the strongest, most efficient shell structure. The path of least resistance. The natural world never has an exact pattern though -- there are always exterior factors. It makes perfect sense, it's not mystical or anything.
> How could Fibonacci have known or seen this pattern?
Uh, he just added the two previous numbers. 0, 1, 1, 2, 3, 5, 8, 13. It's pretty simple really.
> "coincidences" in our day-to-day experience?
What coincidences? Repetitive growth is natural, the path of least resistance. IMO numerology is just the failure of inductive reasoning. Nothing magical.
I'm guess you saw the Touch pilot with Kiefer Sutherland? These themes were really popular in indie films in the 90's. Pi directed by Daren Arronofsky in particular deals with this stuff pretty heavy-handedly.
Pi (1998)
http://www.imdb.com/title/tt0138704/
I also saw a BBC documentary recently on math in the natural world you might like. It was pretty good, scientific.
BBC The Code
http://www.bbc.co.uk/tv/features/code/

The Fibonacci number appears in any spiral pattern on plants - such as the arrangement of seeds in a sunflower or a pinecone, or the petals in a budding flower.
I'm reading now that Fibonacci is present in spiral design such as the inner ear of mammals and the Nautilus shell, and the relative lengths of the human phalanges also follow Fibonacci's first numbers.

If you are taking the sequence as 1, 1, 2, 3, 5, 8, 13, ......... with the first term 1, the second term 1, the third term 2, and so on, it appears that if the terms you multiply are odd numbered terms (subtracting the square of an even term) the result is +1.
If the terms you are multiplying are even numbered terms (subtracting the square of an odd term) the result is - 1.
This is based on inductive reasoning only, and in no way constitutes a proof.

there are a lot of applications of that Fibonacci sequence in mathematics, one of my favorites is the binomial theorem, the pascal's triangle in which that sequence is related. It's hard to enumerate everything, but you could read some articles about it and you'll be amazed on how it is used. About the 144 thing, im not sure if it has something to do with the 12x12, i just know it's the sum of 55 and 89 that's why it is there. ^_^

The Fibonacci has a number of beautiful mathematical properties in geometry and can be found in famous monuments like the angle of the egypthian pyramides, Stonehenge, churges and monasteries - although neither fractional nor irrational numbers were not known until the mid-ages.
http://www.antifool.com/cc/?c=fibonacci
Leonardo Fibonacci was born in Pisa in the 12th century. He was a merchant and customs officer of the time, traveling widely in North Africa. He was also one of the first Europeans to learn about the Arabic numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 and to persuade other people to use them; before then everybody counted in 12's. Leonardo was trying to find a way of modelling the population of rabbits.
Let us suppose that any new pair of rabbits produces one pair in the next breeding season and one in the season after that, and then they die. This means that the total number of new pairs in a given season is equal to the number of new pairs born in the previous season, plus the number born in the season before that. So to find the next number in the sequence you add together the last number and the one before it. Starting with one pair of rabbits, you can easily generate the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... - the population of rabbits grows very quickly - actually exponentially fast.
The surprising thing about Fibonacci's sequence is that it turns out to occur in many different places in nature. The way in which the spiral patterns of sunflower seeds and pine cones grow is described by the sequence, and it is common for the number of petals on a flower to be a Fibonacci number. Four-leaved clovers are rarer than five-leaved ones because five is in Fibonacci's sequence and four isn't.

All sequences are never ending, especially if they rely on previous terms.

The golden ratio is a mathematical constant that occurs when the ratio between two numbers is equal to the ratio between the sum of the two numbers and the larger of the two numbers.
If a > b. The ratio of a : b is equal to the ratio of a+b : a. This occurs when a is 1.61803398874989 times bigger than b.
Therefore if a = 1.61803398874989b, then a+b = 1.61803398874989a
The golden ratio is connected to the fibonnacci sequence. I'm not sure how. The fibonacci sequence is a sequence of numbers formed by adding together the two previous numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
A lot of people think that people find people attractive when different parts of their body are in the golden ratio. I'm pretty sure this is bullshit. I'm not aware of any research that has suggested a correlation betwen the golden ratio and attractiveness. However the golden ratio does appear in a lot of places in nature. Some research suggests there are structures on an atomic scale that exhibit the golden ratio.

seems pretty easy
lts look at the basics again
That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711
in this example 0 plus 1 is 1, 1 plus 1 is 2 , 1 plus 2 is 3, 2 plus 3 is 5.
no I wont do you home work for you you do it there is your answer if your THINK about it

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