Consider the Fibonacci sequence $\{a_n\}$

Question

Consider the Fibonacci sequence $\{a_{n}\}$ Use mathematical induction to prove that $a_{n+1}a_{n-1}=(a_{n})^{2}+(-1)^{n}$

So far, I have tested the base case $n=1$ which is true. I am stuck on the inductive step where I plug in $k=n+1$.

$a_{n+2}a_{n}=(a_{n+1})^{2}+(-1)^{n+1}$

$a_{n+2}a_{n}=(a_{n+1})^{2}-(-1)^{n}$

I am unsure what the next step to take is.

Answer 1

Recall that: $a_1=1$,$a_2=1$,$a_3=2$ .

For $n=2$: $$ a_3a_1=a_2^2+(-1)^2=2 $$

We assume the hypothesis valid for $n=k$, such that: $$ a_{k+1}a_{k-1}=a_k^2+(-1)^k $$

Let's find the value of $a_{k+2}a_{k}$. We can use the Fibonacci recursion: $a_{k+2}=a_{k+1}+a_k$. Therefore: $$ a_{k+2}a_{k}=(a_{k+1}+a_k)a_k=a_{k+1}a_{k}+a_k^2 $$ From the hypothesis: $$ a_{k+1}a_{k-1}=a_k^2+(-1)^k \Rightarrow a_k^2 = a_{k+1}a_{k-1}-(-1)^k $$

Thus: $$ a_{k+2}a_{k}=(a_{k+1}+a_k)a_k=a_{k+1}a_{k}+a_{k+1}a_{k-1}-(-1)^k $$ $$ a_{k+2}a_{k}=a_{k+1}a_{k}+a_{k+1}a_{k-1}+(-1)^{k+1}=a_{k+1}(a_{k}+a_{k-1})+(-1)^{k+1} $$ Apply Fibonacci recursion again to find: $$ a_{k+2}a_{k}=a_{k+1}^2+(-1)^{k+1} $$

Answer 2

For $n=1$, $~~a_{2}a_{0}-a_1^2=-1$, so the statement is true for $n=1$. Let, the statement $P(n):$ $ a_{n+1}a_{n-1}-a_n^2=(-1)^n$ is true for $n=m$. We need to show that $P(m+1)$ is true. $$a_{m+2}a_{m}-a_{m+1}^2=(-1)^{m+1} $$

Using recurrence relation on $a_{m+2}=a_{m+1}+a_m$ and $a_m=a_{m+1}-a_{m-1}$, $$\require{cancel}{a_{m+2}a_m-a_{m+1}^2=(a_{m+1}+a_{m})(a_{m+1}-a_{m-1})-a_{m+1}^2\\ = \cancel{a_{m+1}^2}+a_ma_{m+1}-a_{m+1}a_{m-1}-a_ma_{m-1}-\cancel{a_{m+1}^2}\\=a_m(a_{m+1}-a_{m-1})-a_{m+1}a_{m-1}= a_m^2-a_{m+1}a_{m-1}=-(a_{m+1}a_{m-1}-a_m^2)=(-1)^{m+1} }$$ as by inductive argument $a_{m+1}a_{m-1}-a_m^2=(-1)^m$. Hence done!

Answer 3

The base hypothesis is obviously true: $0\cdot1-1^2=(-1)^1$.

Now the given formula hints to establish $$F_{n+1}F_{n-1}-F_n^2=-(F_{n}F_{n-2}-F_{n-1}^2).$$

If we expand $F_{n+1}$ we get $$F_{n}F_{n-1}+F_{n-1}^2-F_n^2=-F_{n}F_{n-2}+F_{n-1}^2.$$

Then regrouping the two terms with $F_n$, $$F_{n}(F_{n-1}-F_n)+F_{n-1}^2=-F_{n}F_{n-2}+F_{n-1}^2$$

and we are done.