Exercise 38 Page 279

One particularly well-known sequence that is defined recursively is the Fibonacci sequence, in which each term is the sum of the two previous terms. It is a well-known sequence whose first few terms are $1,1,2,3,5,8\dots$ As shown, the first two terms are $1$ and $1$ and each subsequent term will be the sum of the previous two terms.

For every term after the second term, the sequence can be expressed by the following recursive rule. This means that, in order to determine the fourth term of the sequence, $a_4$, one must have the terms $a_3$ and $a_2$. The sequence is named after the Italian mathematician Leonardo Fibonacci, who used it to describe how pairs of rabbits increases as they multiply under certain conditions. Sometimes, the first two numbers are defined as $a_1=0$ and $a_2=1.$ This, however, does not change the sequence in any way other than by increasing the index of every term by one compared to the previously mentioned definition.