Fibonacci Trick
from the MATHEMATICS TEACHER May 1975
This trick works well with a blackboard and a class of students.
Tell your students that they can pick any two single-digit numbers and then you will give them
the sum of a sequence of ten terms that they generate.
The sequence is generated by taking the first two numbers and adding them together to get the third
term. Then add the second and third terms to get the fourth term. Then add the
third and fourth terms to get the fifth term, and so on until you have ten terms.
When the students have written the seventh term, it is time for you to "predict" what the sum will
be. Do this by multiplying in your head 11 times the seventh term.
Here is an example:
Let's say the students picked 2 and 4.
(1) Write down 2 for the first term.
(2) Write down 4 for the second term.
(3) Write down 6 for the third term
(the sum of 2 and 4).
(4) Write down 10 for the fourth term
(the sum of 4 + 6).
(5) Write down 16 for the fifth term.
(6) Write down 26 for the sixth term.
(7) Write down 42 for the seventh term.
Here is where you should stop and write down the sum.
Mentally multiply 11 x 42 to get 462;
Then write 462 at the bottom of the column of numbers
allowing room for three more numbers (terms 8, 9, and 10).
(8) Write down 68 for the eighth term.
(9) Write down 110 for the ninth term.
(10) Write down 178 for the tenth term.
Then have your students find the sum of the ten numbers!!!
Why does this work?
Use some algebra to prove it.
Let x = the first number and y = the second number.
Here are the ten terms:
(1) x
(2) y
(3) x + y
(4) x + 2y
(5) 2x + 3y
(6) 3x + 5y
(7) 5x +8y
(8) 8x +13y
(9) 13x +21y
(10 21x + 34y
The sum of these ten terms is 55x + 88y
which factors into 11 (5x + 8y).
Now look at the 7th term -->
it is 5x + 8y!