It is a description of how to apply Mathematical Induction in proving theorem or statement. Application of mathematical Induction : F 2n+1 - Fn Fn+2 = (-1) n

Application of Mathematical Induction



Written By :- Thokchom Sarojkumar Sinha

Mathematical Induction :- If a statement is true in the first case and
if it is true for all the cases preceding a given one then the statement
is true for this case.

Fibonacci Numbers :- The Fibonacci numbers are numbers that has the
following properties.

If Fn represents the n th Fibonacci number,



F1 = 1, F2 =1, F3 =2, F4=3, F5 = 5 etc.

≥ 3 by using the relation

Fn= Fn-1 + Fn-2 for n ≥ 3

Application of mathematical Induction

To prove

F 2n+1 - Fn Fn+2 = (-1) n

Solution : Before proving it, try to understand the following steps to
be perform.

Step I : Prove the theorem for n =1

Step II : Prove that if the theorem is true for n = k, then it is true
for n = k + 1

Step III : Conclude that the statement is true for all cases.









Proof the theorem by using the above steps :



Step I.:-

For n = 1, we get,



L.H.S. = F 2 1+1 -F1 F1+2 = F22 -F1F3 = 12 -1.2 = 1-2 = -1



R.H.S. = (-1) 1 = -1

Hence the statement is true for n = 1,

Step II. :- Assuming that the statement is true for n = k



i.e. F 2 k+1 -Fk Fk+2 = (-1)k



Then for n = k+1, we get,

F 2(k+1)+1 -Fk+1 F(k+1) +2

= F 2k+2 -Fk+1 Fk+3

= (F k+1 + F k)2 - F k+1 (F k+2 + F k+1)

Since F k+2 = F k+1+ Fk

= F2k+1 + 2F k+1 F k + F2 k + F2 k -F k+1 F k+2 -F2k+1

= F k (Fk+1 + Fk) + Fk Fk+1 -Fk+1 Fk+2

= Fk Fk+2 + Fk Fk+1 -Fk+1 (Fk+1 + Fk)

Since Fk+2 = Fk+1 + Fk

Therefore,

F2(k+1)+1 - F k+1 F(k+1) +2

= Fk Fk+2 + Fk Fk+1 - F2k+1 - FkF k+1

= Fk Fk+2 - F2k+1

= - (F2k+1 -Fk Fk+2)

= -(-1) k By Hypothesis F2k+1-FkFk+1 = (-1)k

= (-1)k+1

Step III : We conclude that the statement is true for all cases.

Since the statement is true for n=1, then it will be true for n = 1+1 =
2. Again it will also be true for n = 2+1 = 3 and so on.

Reference :-

1. Number theory- George E. Andrews.