Fibonacci sequence#

Boaler (2015) suggests it is the best known of all patterns. Named after an Italian mathematician who published on the sequence in 1202. However, earlier evidence of the sequence has been found in Indian mathematics as far back as 200 b.c. In fact, it was used in the meter of Sanskrit poetry.

The sequence can be said to start with 0 and 1, or with 1 and 1. The sequence is then generated by adding the last two numbers in the sequence to get the next number.

\[0, 1, 1, 2, 3, 4, 8, 13, 21, 34, 55...\]

Dividing each number by the previous number in the sequence gives a sequence of numbers that approaches the golden ratio.

Mathematical representation#

\[ \displaylines{F_0 = 0, F_1 = 1, \\ \text{and} \ \\ F_n = F_n-1 + F_n-2 \\ \text{ for } n > 1.} \]

References#

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons, Incorporated.