1. To Built the Fibonacci series the first thing to do is place the first term
that is 1.
The next step is determinate the second element of the series, for this
we add the first term of the series with the previous number of this, that is
to say, if the first element is one then the previous number of this is zero,
therefore the second term of the series is 1 + 0 = 1.
In the same way to find the third element of the series we add the second
element with the first element of the series, that is to say, since the second
element of the series is 1 and the first element of the series is 1 the third
element of the series is 1 + 1 = 2.
In the same way to find the fourth element of the series we add the third
element of the series with second element of the series, since the third element
of the series is 2 and the second element is 1, therfore the fourth element is
2 + 1 = 3 and so on.
T erm
N ext term
F irst 1
Second term = 1 + 0 = 1
Second
Second 1
T hird term =
1
F irst
+ 1 =2
T hird
T hird
Second
2 F ourth term = 2 +
F ourth
F ourth 3
F if th term =
3
5
.
.
.
=3
+ 2 =5
F if th
F if th
.
.
.
1
T hird
Sixth term = 5 +
.
.
.
F ourth
3
=8
Using this we can write the first ten terms the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
2. Sum of the first ten terms of the Fibonacci sequence is
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143.
Division of the sum between 11, 143 / 11 = 13.
Since the division is exact the sum of the first ten terms of the Fibonacci
sequence it is multiple of 11.
3. Let’s choose 3 and 4, following the procedure given in problem 1 we
2
have:
T erm
F irst
N ext term
3
Second
Second
4
T hird term =
4
F irst
+ 3 =7
T hird
T hird
7
F ourth term = 7 +
F ourth
F ourth 11
Second
4
= 11
T hird
F if th term = 11 + 7 = 18
F if th
F ourth
F if th 18 Sixth term = 18 + 11 = 29
.
.
.
.
.
.
.
.
.
3, 4, 7, 11, 18, 29, 47, 76, 123, 199.
4. The sum of the sequence previous is given by:
3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123 + 199 = 517.
The division of the sum by 11 is: 517 / 11 = 47.
Since the division is exact the sum of the first ten term of the Fibonacci-
like sequence is multiple of 11.
5. Hypothesis: The sum of the first ten term of any Fibonacci-like is
multiple of 11.
6. Let’s choose 7 and 8, following the procedure given in problem 1 we
have:
7, 8, 15, 23, 38, 61, 99, 160, 259, 419.
The sum of the first ten term of the sequence is:
7 + 8 + 15 + 23 + 38 + 99 + 160 + 259 + 419 = 1089.
The division of the sum by 11, 1089 / 11 = 99.
Since the sum of the first ten term of the sequence is multiple of 11 the
hypothesis apply to this sequence (remember if the division of the sum of the
first ten term of the sequence between 11 is exact, the sum of the first ten
term is multiple of 11)