Stuck on some proofs

can anyone help me with the proofs:

[tex]1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots=\frac{\pi^2}{6}[/tex]

if [tex]F_i[/tex] is the ith Fibonacci number, then

[tex]F_1+F_2+F_3+\ldots+F_n=F_{n+2}-1[/tex]

[tex]F_2+F_4+F_6+\ldots+F_{2n}=F_{2n+1}-1[/tex]

[tex]F_1+F_3+F_5+\ldots+F_{2n-1}=F_{2n}[/tex]

[tex]F_1^2+F_2^2+F_3^2+\ldots+F_n^2=F_nF_{n+1}[/tex]

I proved the last four using induction. But how can i prove them without using induction?
 

lurflurf

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murshid_islam said:
can anyone help me with the proofs:

[tex]1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots=\frac{\pi^2}{6}[/tex]

if [tex]F_i[/tex] is the ith Fibonacci number, then

[tex]F_1+F_2+F_3+\ldots+F_n=F_{n+2}-1[/tex]

[tex]F_2+F_4+F_6+\ldots+F_{2n}=F_{2n+1}-1[/tex]

[tex]F_1+F_3+F_5+\ldots+F_{2n-1}=F_{2n}[/tex]

[tex]F_1^2+F_2^2+F_3^2+\ldots+F_n^2=F_nF_{n+1}[/tex]

I proved the last four using induction. But how can i prove them without using induction?
euler found your sum zeta(2) by writing
sin(x)=x(1-x^2/(k pi)^2)(1-x^4/(k pi)^4)(1-x^6/(k pi)^6)...
and expanding the product and setting it equal x-x^3/3!+x^5/5!+...
It is often done as a routine exercise with fourier series.
There are many ways to prove it.
see
http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html
for the other 4 whats wrong with induction?
Fn=(a^n-b^n)/(a-b)
where a=.5(1+sqrt(5)) b=.5(1-sqrt(5))
so you could express those sums as geometric series among other methods
see
http://mathworld.wolfram.com/FibonacciNumber.html
 
lurflurf said:
for the other 4 whats wrong with induction?
there's nothing wrong with induction. but i wanted get the right hand sides from the left hand sides (from scratch), if you know what i mean.

anyway, thanks very much for the help. and additional information is always welcome.
 

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