Let displaystyle{F}_{{i}} be in the displaystyle{i}^{{{t}{h}}} Fibonacc number, and let n be ary positive eteger displaystylege{3}Prove thatdisplaystyle{F}_{{n}}=frac{1}{{4}}{left({F}_{{{n}-{2}}}+{F}_{{n}}+{F}_{{{n}+{2}}}right)}

defazajx 2021-01-19 Answered

Let Fi be in the ith Fibonacc number, and let n be any positive integer 3
Prove that
Fn=14(Fn2+Fn+Fn+2)

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Answered 2021-01-20 Author has 98 answers

Let us first recall a definition of nth Fibonacci number
Fn=Fn1+Fn2,for n2
Now we have to show
Fn=14(Fn2+Fn+Fn+2),for n3
Now starting from right hand side we get
14(Fn2+Fn+Fn+2)=14(Fn2+Fn+(Fn+1Fn))[Fn+2=Fn+1+Fn]
=14(Fn2+2Fn+Fn+1)[Fn+2=Fn+1+Fn]
=14(Fn2+2Fn+(FnFn1))[Fn+1=Fn+Fn1]
=14((Fn2+Fn1)3Fn)
=14(4Fn)[Fn=Fn1+Fn2]
=Fn
Hence the proved

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