- #1
murshid_islam
- 457
- 19
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?
[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?