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58 At Right Angles | Vol. 3, No. 1, March 2014

The Fibonacci Puzzle

The Fibonacci puzzle was posed by Leonardo of Pisa, also known as

Fibonacci (hence the name of the sequence). The puzzle describes

the growth of an idealized rabbit population. A newly born pair of

rabbits comprising a male and a female rabbit is put in the field and

are able to mate at the age of one month. At the end of the second

month the female rabbit produces a pair of rabbits (again a male

and a female). Rabbits never die and every mating pair always

produces a new pair every month from the second month on. How

many pairs will there be in one year?

Let us try to find out the number of pairs of rabbits at the end of

every month starting from the first month. To begin with there is

one pair of rabbits. At the end of the first month, they mate, but

there is still only 1 pair. At the end of the second month the female

produces a new pair, so now there are 2 pairs of rabbits in the

Exploring

Fibonacci Numbers

Jonaki Ghosh

techspace

Keywords: sequence, Fibonacci, golden ratio, investigation, Excel

sequence of numbers known as the Fibonacci sequence.

The exploration of the sequence can lead to an

absorbing classroom activity for students at the middle

school and secondary school level. Students can explore

many patterns within the sequence using a spreadsheet

like MS Excel and the observations can lead to an

enriching discussion in the classroom. ## Page 2 of 6

Vol. 3, No. 1, March 2014 | At Right Angles 59

field. At the end of the third month, the female of the original pair produces a second pair, making 3 pairs

altogether in the field. Remember that the second pair which was born at the end of the second month is

only able to mate at the end of the third month. At the end of the fourth month, the female of the original

pair has produced yet another new pair and the female born at the end of the second month produces her

first pair, making 5 pairs in all. This process continues. To obtain the number of pairs at the end of any

given month, say n, we need to add the number of pairs at the end of month n−1 and the number of pairs

at the end of month n−2.

Hence the Fibonacci sequence can be written in the form of the recurrence relation

Fn = Fn − 1 + Fn − 2

The first two terms of the sequence are 1 and 1. So if F1= F2= 1, the recurrence relation can also be written as

Fn + 2 = Fn + 1 + Fn .

Sometimes F0 is taken to be 0. Here are the first fifteen Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,

144, 233, 377, 610.

Can you guess how large the 50th Fibonacci number will be? One can continue the process of adding pairs

of consecutive terms to find the next term, but after a while it can get quite cumbersome. Let us take the

help of a spreadsheet to generate the Fibonacci sequence and find its 50th term.

Generating the Fibonacci sequence on Excel: The following steps will help you obtain the sequence on

Excel.

Step 1: The first step is to create a column of integers

from 1 to 50 (in column A). This may be achieved in

the following manner.

Click on cell A2 and enter 1. Then enter = A2 + 1

in cell A3 and drag cell A3 till A51. This will create a

column of numbers 1 to 50.

Step 2: Enter 1 in cells B2 and B3. In cell B4, enter =

B3 + B2 and press enter. A double click on the corner

of cell B4 will generate the Fibonacci sequence as

shown in Figure 1. As you scroll down the sequence

in column B, you will notice that the column width is

too small to accommodate the numbers. For example

the 40th Fibonacci number appears as 1.02E + 08.

This means that Excel has approximated the number

and the number is close to 1.02 × 108

. To get the

actual terms of the sequence beyond the 40th term,

the column width needs to be increased. This can be

done by taking the cursor to the end of the column

(were the columns are named as A, B, C etc) and

dragging it to the required width.

Note that the 50th Fibonacci number is 12586269025.

Clearly, this is a very fast growing sequence. See

Figure 2. Figure 1: Generating the Fibonacci sequence on Excel. ## Page 3 of 6

60 At Right Angles | Vol. 3, No. 1, March 2014

Step 3: We can also draw the graph of the sequence. For this, select column B by clicking on B, go to Insert

on the toolbar, select chart and then line. See Figure 3.

As you already know,the first 15 Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,

377, 610. Let us see what happens when we take the ratios of consecutive terms Fn+1

Fn

of the Fibonacci

sequence. They are:

.......

Let us compute the ratios on Excel as follows. In cell C3 of

column C, enter = B3 / B2 and double click on the corner

of the cell. You will observe that after a certain number of

terms the ratios become steady at 1.618034.

A natural question now arises whether this value (that

is, 1.618034) will remain the same if we change the

initial values of the Fibonacci sequence (which are 1 and

1). Let us try to investigate by taking different starting

values. Change the values in cells B2 and B3 in the Excel

sheet to 4 and 7. Observe that the ratios of successive

terms still approach 1.618034 (shown on Figure 5).

You may investigate by taking different starting values.

You will observe that the ratios of successive terms still

approach 1.618034. A graph of the ratios of successive

terms (obtained by selecting column C) reveals this

behavior of the sequence. See Figures 5 and 6. This is

indeed an interesting observation but we need to find a

mathematical explanation for it.

Figure 2: A part of the Excel sheet

showing the 39th to 50th terms of

the Fibonacci sequence. Figure 3: Graphing the Fibonacci sequence on Excel.

Figure 4: The ratios of consecutive terms of the

Fibonacci sequence become steady at 1.618034

(in column C) 