In: Advanced Math
For the Fibonacci sequence, prove the formula u2n+1 = un un+2 + (-1)n
The Febonacci sequence is given by where .
We have to prove the formula
We shall prove this formula by using Strong Mathematical Induction on the set of natural numbers
for n = 1, the formula is
Also from the febonacci sequence we have
since
hence the formula is true for n = 1.
suppose the formula is true for n = 1,2,3,...,k
that is ........(2)
now take n = k+1,
from equation (1) we have
using equation (2)
using equation (1)
again using equation (1)
as
this is the formula for n = k+1
Hence the formula is true for n = k+1
Thus the formula is true for n = 1 and if we assume that it is true for n = 1,2,3,...,k, then it is true for n = k+1
therefore by using Mathematical induction we conclude that the formula is true for every .
Hence for the Febonacci sequence