Question

In: Advanced Math

For the Fibonacci sequence, prove the formula u2n+1 = un un+2 + (-1)n

For the Fibonacci sequence, prove the formula u²_n+1 = u_n u_n+2 + (-1)ⁿ

Expert Solution

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

Colby Messinger answered 3 years ago

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