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  4. If F is function such that F (0) = 2, F (1) = 3, F (x+2) = 2F (x) - F (x+1) for x ge 0, then F (5) is equal to

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

VITEEEVITEEE 2010Relations and Functions

Solution:

We have,
$F(x+2)=2 F(x)-F(x+1)$ ... (i)
Putting $x=0$, we get
$F(2)=2 F(0)-F(1)$
$\Rightarrow F(2)=2(2)-3$
$\{\because F(0)=2, F(1)=3\}$
$\Rightarrow F(2)=4-3$
$\Rightarrow F(2)=1$
Putting $x=1$, in Eq. (i), we get
$F(3) =2 F(1)-F(2)$
$=2(3)-1\{\because F(1)=3, F(2)=1\}$
$\Rightarrow F(3) =5$
Putting $x=2$, in Eq. (i), we get
$F(4) =2 F(2)-F(3)$
$=2(1)-5 \{\because F(2)=1, F(3)=5\}$
$F(4) =-3$
Putting $x=3$, in Eq. (i), we get
$F(5) =2 F(3)-F(4)$
$=2(5)+3\{\because F(3)=5, F(4)=-3\}$
$\Rightarrow F(5) =13$