So, we are discussingthe recursion and last
lecture we have learned how actually we candefine
recursively the sequences functions and the
sets . Today we will see in detailswith some
examples that how recursively definedsequences
can beformed andwhat are their applications.
First we see the Recursively DefinedSequences
. We introduce an integer sequencewhich ispopularly
used incombinatorics and in graph theory it
is called the Fibonacci sequence . So, you
introduce called Fibonacci sequence . So,
first we see how these Fibonacci sequence
are recursively defined the recursive definition
of .
So, as wediscussed last day andwe have defined
recursively defined sequencesfunctions and
sets; that with some initial values and we
can give some rule so that we get a particular
term in the sequence from the previous term
or previous set of terms . So, Fibonacci sequence
it is defined that its initial condition the
initial condition 
are say F 0 equal to 0 . F 1 equal to 1 where
F represents the Fibonacci sequence and the
recursive definitions are or you can tell
recursive rule; that F n equal to F n minus
1 plus F n minus 2.
That means, the nth term of the sequence 
can be found by adding the previous 2 terms
. So, how we can find the some of the terms
using this recursive rule. So, since F 0 and
F 1 are given so if we find that find F 2
F 3 F 4 F 5 F 6. Then I can apply the recursive
rule as well as we take the initial values.
So, F 2 is F 1 plus F 0. And F 1 is given
1 F 0 is 0 so this is 1 . Now F 3 is F 2 plus
F 1 just now we got F 2 equal to 1, and F
1 has the initial condition is given as 1
so it is 2 . Similarly I get F 3 F 4 already
you will got F 3 so F 4 is F 3 plus F 2 and
F 3 is 2 F 2 is 1 so this is 3 . We got F
4 is F 3 plus F 2 and that is 3 plus 2 is
5 .
Then F 5 is F 4 plus F 3 is 5 plus 3 is 8,
then F 6 is F 5 plus F 4 is 8 plus 5 equal
to 13. And if I continue in this way we will
be getting the values of the different terms
of the sequences . So, in this way we can
define the sequence recursively we can define
a sequence. Now we can prove this sequence;
that means, the recursive definition of Fibonacci
sequence in conjunction with the mathematical
induction we canprove we can establish the
some of the interesting properties using theFibonacci
sequences. And which is number of application
in combinatorics as well as graph theory and
other as branches of mathematics .
So, the recursive using the recursive definition
of 
of the of the Fibonacci sequence 
in conjunction with the mathematical induction
. 
We can establish a number of properties 
of these sequences of these sequences . One
very simpleexample we see so we see a propertyP
1 say it the property tells the sum of the
squares of the terms of the Fibonacci sequence
.
So, just now we have seen the terms are F
0 F 1 F 2 F 3 like that the Fibonacci sequence.
And we have a initial condition we have seen
that F 0 equal to 0, F 1 equal to 1. And using
the recursive rules we have found out that
F 2 equal to 1 F 3 equal to 2 F 4 equal to
3 like that already we have seen . Now we
we want to see that; what is the value of
the or the sum of the squares of the terms
of Fibonacci sequence . So, if I write the
say for I equal to 1, then it is sum of squares
is F 0 square plus F 1 square . And this is
0 square plus 1 square is 1 and say I am writing
this is 1 into 1.
Now, if it is I equal to 2 if I equal to 2
then sum becomes F 0 square plus F 1 square
plus F 2 square. And this becomes 0 square
plus 1 square plus F 2 is also 1. So, again
1 square that is 2 and this becomes 1 into
2. Say I equal to 3 so similarly I can write
F 0 square plus F 1 square plus F 2 square
plus F 3 square is 0 square plus 1 square
plus 1 square plus my F 3 is 2 so 2 square.
So, this becomes 6 is 2 into 3 .
Now, if I continue in this way F 2 square
plus F 3 squared plus F 4 square again this
becomes 0 square plus 1 square plus 1 square
plus 2 square plus F 4 is 3. So, 3 square
and this is 9 plus 6 this is 15 this becomes
3 into 5 . Now if we observe the results or
the sum of squares for different values of
I. What we see that we can write that it isfor
I equal to 1 1 into 1 so this becomes F 1
into F 2 because my F 1 and F 2 both are 1
.
Now, my F 2 F 3 is 2 so I can write. So, I
can write this thing as F 1 into F 2 I can
write this thing as the F 2 into F 3. I can
write this thing as the F 3 into F 4. This
as the F 4 into F 5 so we give we get apattern
. So, we can frame the formula that we can
write that the conjecture or the result is
that we can write the conjecture that for
all n belongs to z plus the positiveset of
positive integers. The sum of squares of the
Fibonacci terms that up to n terms if I write
this is F i square and that becomes F n into
F n plus 1 . So, this is first we see the
property that what will be the sum of squares
of in Fibonacci terms and we get this formula.
Now we can prove now we can prove this result
by using the recursive definition and the
principle of mathematical induction . 
As in the last lecture we have described that
recursive definition is closely related with
the mathematical induction. Because the recursive
rule that is framed from the concept of mathematical
induction . 
Now how we can prove that thing.
So, we will be using the mathematical induction.
So, if I get the proof 
so first thing is that as the we have want
to apply mathematical induction. So, we have
to write the basis step . So, for basis step
we that for n equal to 1 we take assume n
equal to n equal to 1 then the conjecture
becomes the n 1 F 1 square or should write
the similar way as it is written that I equal
to 0 to 1 .
F I square equal to F 0 square plus F 1 square
and from the initial conditions this is 0
square plus 1 square is 1 and that I can write
that 1 into 1 is F 1 into F 2. Since we know
that F 2 is also 1, since F 1 equal to 1 from
initial condition initial value or condition
we can write . And F 2 equal to 1 so my conjecture
is true for the basis step n equal to 1.
Now, we see the how we can prove the recursive
step now to prove the recursive step we use
mathematical induction . So, we assume that
the conjecture is true for I equal to n; that
means, for I equal to 0 to n F I square equal
to F 0 square plus F 1 square plus F 2 square
plus F n square this equal to F n into F n
plus one.
Now, if we can show that this conjecture is
true assuming that it is true for I equal
to n the conjecture is true for I equal to
n plus one. So, for I equal to n plus one
we have to show that conjecture is true if
we can show then 0 to n plus 1 . And this
is F i square so I can write this is F 0 square
plus F 1 square plus F 2 square F n square
plus F n plus 1 whole square F n plus 1 . Now
this is from F 0 square to F n square I can
replace by this summation I equal to 0 to
n F I square plus F n plus 1 square.
Now, already according to that principal of
mathematical induction it is the conjecture
is true for I equal to n so I can write this
is F n into F n plus 1. And the next term
is F n plus 1 whole square. So now, if I take
F n plus 1 comma then this becomes F n plus
F n plus 1 is F n plus 1 into F n plus F n
plus 1 is according to the definition of Fibonacci
sequence this becomes n plus 2 . So, here
I use the recursive definition of here we
use the recursive definition of of Fibonacci
sequence and we use the principle of mathematical
induction .
So, this is F n plus 1 F is for I equal to
n plus 1 it is true it is true . So, I can
write that the property for for n belongs
to for all n belongs to the set of positive
integers that summation I equal to 0 2, n
F I square is F n into F n plus 1 this we
can the proof . And in this way in this way
I can prove many such properties forthis Fibonacci
sequence. Nowwe observe or some the importance
of the basis step or the initial condition
.
So, how we define that Fibonacci sequence?
We define the Fibonacci recursive definition
that we have given the Fibonacci sequence
the recursive definition of Fibonacci sequence
that with the initial condition F 0 equal
to 0, F 1 equal to 1. The sequence is F n
is F n minus 1 plus F n minus 2 . Now if we
change the only the basis step so this is
my initial conditions or the basis step I
can write this is my this is my basis step
or my initial condition of the sequence .
So, if I just change this the even if the
rule is recursive rule is same then it will
generate some new sequences so this is my
recursive step . Now one such new sequence
we call that are the Lucas sequence we see
that Lucas sequence 
even if the recursive steps are same. So,
here the recursive 
rule or step recursive rule or recursive step
is same as that of Fibonacci sequence. So
that means, if we can write that represent
this thing as a L, Lucas sequence as L then
L n is L n minus 1 plus L n minus 2 .
But we change the basis step basis step is
changed we write the basis step is changed
or I should tell that initial conditions are
changed initial conditions are changed . So,
initial conditions we write that L 0 equal
to 2 and L 1 is same as that of Fibonacci
F 1 is 1 . Now if we see the sequences we
see the sequences and ok we write this is
my 
basis step say I can I can find out that L
2 which is 0 plus L 1 this is 2 plus 1 equal
to 3. L 3 is L 1 plus L 2 is L 1 is one L
2 is 3 so this is 4.
L 4 is L 3 plus L 2 plus L 3 so this is 3
plus 4 is 7 . So, we see that terms are even
the recursive rules are same as that of Fibonacci
sequence, but the terms are different. Here
L 2 is 3 L 3 is 4 L 4 is 7 and the terms are
different. So, if we can if we prepare the
list we see the hm we can compare the sequences;
the Fibonacci sequences sequence and Lucas
sequence .
You can if we compare the values we see that
if we can put n 
we give L n and we get give the F n and we
see the different values of n. Say for n equal
to 0 1 2 3 4 we take some more values 5 6
7 . And we can see L n is 2 because it is
given the initial conditions then 2 plus 1
3, 3 plus 1 4, 4 plus 3 7, 7 plus 4 11, 11
plus 7 18, 18 plus 29 . And here the initial
conditions are different for Fibonacci it
is 0 and 1. So, it is 1 1 plus 1 2 it is 3
then it is 5 then it is 8, 13 like that. So,
what we see that that this is my Lucas number
and this is my Fibonacci number .
See even the so conclusion is even the recursive
rule or the recursive definition is same;
same since the initial condition or initial
terms of the sequences 
are different they generate different sequences
. Now since the 
recursive rule is same only the initial values
are different . So, there are there is a close
relation between the Lucas numbers and the
Fibonacci numbers. So, we can see some; one
property that again like.
We have seen earlier only the Fibonacci property.
So, one property we have seen that where we
give either relation this gives some relation
between Fibonacci sequence 
and Lucas sequence . 
I write one such property as the see if I
take the sum of Fibonacci numbers I I take
Lucas number is i equal to 
I take a simple property that I can I can
write the Lucas numbersay nth term of the
Lucas sequence L n I can write it is F n minus
one plus F n plus 1 .
So, if I see the list of sequence quickly
we can see that my basis step 
is L 0 equal to 2, L 1 equal to 1 so 
I can write and the recursive definition if
I give recursive definition 
of Lucas number. We know that L n is L n minus
1 plus L n minus 2 . So, I can write that
it is I I can find that L 2 since L 2 is L
0 plus L 1. And we have seen already the numbers
are 3 etcetera ok. And I can write 3 as 1
plus 2 and if you remember that this is F
1 I can write F 1 plus F 3 is 2.
Because if we remember the Fibonacci F 0 equal
to 0 F 1 equal to 1 F 2 equal to 1, F 3 equal
to 2 then F 4 equal to 3 like that. So, what
is my L 3 L 3 is L 1 plus L 2 and L 1 is 1
L 2 is 3. So, this is 4 and I can write that
this 
is 4 is 3 plus 1. That means,F 2 plus F 4.
If I continue in this way L 4 is L 2 plus
L 3 and this is my L 2 is 3 plus L 3 is 4
this is 7 and I can write that this is F 3
2 plus 5 F 3 plus F 5.
So, I can write this is 2 plus 5 this is F
2 is 1 plus 3 this is also 1 plus F 1 is 1
F 3 is 2. So, I can see that the property
holds. And the similar way we can write that.
So, we can see that property holds for that
L n is L n is F n minus 1 plus F n minus 2.
And the similar way we can we can give the
proof also .
We can see that for basis step if it is for
n equal to 1 let my L 1 or for n equal to
2 if I see that n equal to 2 L 2 is L 0 plus
L 1 is 2 plus 1 equal to 3 equal to my F 1
plus F 3 1 plus 2 . So, it is true and for
recursive step assume for assume it is true
for n assume it is true for n. So, L n equal
to F n minus 1 plus F n minus 2 true. So,
for i equal to n plus 1 what I can show L
n plus 1 is L n plus L n minus 1.
Now, I can apply the strong form of mathematical
induction here we can use strong form of mathematical
induction. So, that for L n L n minus 1 for
both we can apply the conjecture and then
for L n it is F n minus 1 plus F n minus 2
and for L n minus 1 it is F n minus 2 plus
F n. So, if I take together the for L n it
is L n isL 0 plus L 1 L minus 1 plus F and
n plus 1 ok . This is L minus 1 plus F n plus
1 for n minus 1 it is n minus 2 and n.
So, this becomes if n minus 1 F n plus F n
plus 2 is F n plus 2 and n plus n minus 2
and n minus 1 and n minus 2 this becomes F
n and this becomes F n plus 2. So, which is
the F n plus 1 minus 1 plus F n plus 1 plus
1 so it is proved . So, we see that how recursively
definedsequencescan be defined and for new
sequences. And how we get the properties;
how we can prove the properties using the
mathematical induction as well as the recursive
definitions of the sequence .
