We are asked how many rabbits there will be after one year. One year is 12 months. This tells us that we should calculate the 12^(th) term of the Fibonacci sequence.
Recursive Rule
We will find f_(12) using the following recursive formula for the Fibonacci sequence.
f_1=1,f_2=1
f_n=f_(n-2)+f_(n-1)forn>2
Let's do it!
n
f_(n-2)
f_(n-1)
f_(n-2)+f_(n-1)
f_n=f_(n-2)+f_(n-1)
1
-
-
-
f_1= 1
2
-
-
-
f_2= 1
3
f_1= 1
f_2= 1
1+ 1= 2
f_3= 2
4
f_2= 1
f_3= 2
1+ 2= 3
f_4= 3
5
f_3= 2
f_4= 3
2+ 3= 5
f_5= 5
6
f_4= 3
f_5= 5
3+ 5= 8
f_6= 8
7
f_5= 5
f_6= 8
5+ 8= 13
f_3= 13
8
f_6= 8
f_7= 13
8+ 13= 21
f_8= 21
9
f_7= 13
f_8= 21
13+ 21= 34
f_9= 34
10
f_8= 21
f_9= 34
21+ 23= 55
f_(10)= 55
11
f_9= 34
f_(10)= 55
34+ 55= 89
f_(11)= 89
12
f_(10)= 55
f_(11)= 89
55+ 89= 144
f_(12)= 144
Therefore, f_(12)=144. This tells us that there will be 144 rabbits after one year.
Exact Rule
Now, we will find f_(12) using the following exact rule for the Fibonacci sequence.
f_n=1/sqrt(5)(1+sqrt(5)/2)^n-1/sqrt(5)(1-sqrt(5)/2)^n , n≥ 1
Let's substitute 12 for n into the formula and find f_(12). We will use a calculator. Since f_(12) is an integer, we will round the answer to the nearest integer.
Therefore, f_(12)=144. This tells us that there will be 144 rabbits after one year.
Extra
Recursive Rule and a Spreadsheet
We will find f_(12) using a recursive rule and a spreadsheet. Let's enter a_1=1 in cell A1, a_2=1 in cell A2, and write =A1+A2 in cell A3. When we hit enter, we will get the following.
Next, in the spreadsheet copy cell A3, highlight cells A4 through A12, and paste.
The spreadsheet in the 12^(th) row shows the number 144. Therefore, f_(12)=144.
