Challenge Level

The Fibonacci sequence is

$1, 1, 2, 3, 5, 8, 13, 21 \ldots $

where each term is the sum of the two terms that go before it (i.e $1+1=2$, $1+2=3$, $2+3=5$ and so on.)

What is the sixth term of the Fibonacci type sequence that starts with $2$ and $38$ as the first two terms?

How many Fibonacci type sequences can you find containing the number $196$ as one of the terms where the sequence starts with two whole numbers $a$ and $b$ with $a< b$?

NOTES AND BACKGROUND

Fibonacci sequences are named after a merchant, one Leonardo of Pisa who had the nickname Fibonacci. On his travels, around 1200 AD, he learnt a lot of mathematics (particularly algebra) from the Arabs.

The Arabs had developed the study of mathematics for about 800 years after the fall of the Greek and Roman civilisations. The story behind the methods in this problem spans this whole period.

Equations in which one seeks whole number solutions, are called Diophantine equations after Diophantus (250 A.D) who developed the method and a special notation for recording it.

For more about the stories of Diophantus and Fibonacci see the History of Maths website .