What is the 100th term in the Fibonacci Sequence?

Question

I need to write a paper about the Fibonacci sequence for Algebra and one of the requirements is to write down the 100th term in the Fibonacci Sequence.

Answer 1

I didn't do the calculations, but I searched on it and found several sites with all sorts of info.

100th term is 354224848179261915075
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Answer 2

You might think you have to write down the first 100
Fibonacci numbers and add them. Not so! Let's find
a shorter way. Recall that the Fibonacci numbers
are defined by
F_1 = 1, F_2 = 1, F_(n+1) = F_n + F_(n-1).
Let's also consider the associated Lucas sequence
defined by
L_1 = 1, L_2 = 3, L_(n+1) = L_n + L_(n-1).

Let's write down the first 13 terms of each sequence.
It will turn out, that from these, I can get
F_25, then L_25, then F_50, L_50 and finally F_100.
L_n F_n
1 1
3 1
4 2
7 3
11 5
18 8
29 13
47 21
76 34
123 55
199 89
322 144
521 233

If you look carefully, you will notice the
following identities.
F_2n = F_n*L_n,
F_(2n+1) = F_n² + F_(n+1)²
and finally
L_n² -5F_n² = 4(-1)^n,
which can all be proved either by induction
or by using Binet's formula. This goes
as follows:

Binet's formula:
Let α = (1 + √5)/2, β = (1 - √5)/2
Then
L_n = α^n + β^n and F_ n = (α^n - β^n)/√5.
Now note that if you multiply L_n and F_n you
get the formula for F_2n. This proves the first
of the identities. You can get the others in the
same way.
So we find F_25 = 75025 (= F_12² + F_13².)
L_25² = 5*75025² -4 =28143753121
L_25 = 167761.
First step accomplished!
Now let's find F_50 and L_50.
F_50 = L_25*F_25. = 12586269025
L_50 = √(5F_50² + 4) = 28143753123
and finally,
F_100 = L_50*F_50 = 354224848179261915075.
There you are!
In your paper, you might mention 2 notorious
unsolved problems regarding the Fibonacci sequence:
1). Are there infinitely many Fibonacci primes?
2). Is the first Fibonacci number divisible by a
prime p ever divisible by p²? This problem
has deep connections with Fermat's last theorem!
Hope that helps a bit!

Answer 3

The answer is 354224848179261915075.

Fibonacci Series, in mathematics, series of numbers in which each member is the sum of the two preceding numbers. For example, a series beginning 0, 1 ... continues as 1, 2, 3, 5, 8, 13, 21, and so forth. The series was discovered by the Italian mathematician Leonardo Fibonacci (circa 1170-c. 1240), also called Leonardo of Pisa. Fibonacci numbers have many interesting properties and are widely used in mathematics. Natural patterns, such as the spiral growth of leaves on some trees, often exhibit the Fibonacci series.

Answer 4

First 100 Fibonacci Numbers

Answer 5

The answer for the 100th term is 354224848179261915075.