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[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?