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  4. If a function F is such that f(0)=2, F(1)=3, F(n+2)=2F(n)-F(n+1) for n≠ 0, then F(5) is equal to

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Q. If a function F is such that $ f(0)=2,\,\,F(1)=3, $ $ F(n+2)=2F(n)-F(n+1) $ for $ n\ne 0, $ then $ F(5) $ is equal to

J & K CETJ & K CET 2003

Solution:

Given that, $ F(0)=2,\,\,F(1)=3, $ $ F(n+2)=2F(n)-F(n+1) $ At $ n=0,\,\,F(0+2)=2F(0)-F(1) $
$ \Rightarrow $ $ F(3)=2(2)-3=1 $ At $ n=1,\,F(1+2)=2F(1)-F(2) $
$ \Rightarrow $ $ F(3)=2(3)-1=5 $ At $ n=2,\,\,F(2+2)=2F(2)-F(3) $
$ \Rightarrow $ $ F(4)=2(1)-5=-3 $ At $ n=3,\,F(3+2)=2F(3)-F(4) $
$=2(5)-(-3) $
$ \Rightarrow $ $ F(5)=13 $