Question

If a $functionF$ is such that $F\left(0\right)=2,F\left(1\right)=3,F(x+2)=2F\left(x\right)-F(x+1)forx\ge 0$, then $F\left(5\right)$ is equal to

• A. $-7$
• B. $-3$
• C. $17$
• D. $13$

Answer 1

The correct option is D

$13$

Explanation for correct option:

Step1. Finding value for$F\left(2\right)$

Given, $F\left(0\right)=2,F\left(1\right)=3$

$F(x+2)=2F\left(x\right)–F(x+1)\dots .\left(i\right)$

Now, put $x=0$ in $\left(\mathrm{i}\right)$

$\begin{array}{rcl}F(0+2)& =& 2F\left(0\right)–F(0+1)\\ & \Rightarrow & F\left(2\right)=2F\left(0\right)–F\left(1\right)\\ & \Rightarrow & F\left(2\right)=2\times 2–3\\ & \Rightarrow & F\left(2\right)=4–3\\ & \Rightarrow & F\left(2\right)=1\end{array}$

Step2. Finding value for $F\left(3\right)$

Put $x=1in\left(i\right)$

$\begin{array}{rcl}F(1+2)& =& 2F\left(1\right)–F(1+1)\\ & \Rightarrow & F\left(3\right)=2F\left(1\right)–F\left(2\right)\\ & \Rightarrow & F\left(3\right)=2\times 3–1\\ & \Rightarrow & F\left(3\right)=6–1\\ & \Rightarrow & F\left(3\right)=5\end{array}$

Step3. Finding value for $F\left(4\right)$

Put $x=2in\left(i\right)$

$\begin{array}{rcl}F(2+2)& =& 2F\left(2\right)–F(2+1)\\ & \Rightarrow & F\left(4\right)=2F\left(2\right)–F\left(3\right)\\ & \Rightarrow & F\left(4\right)=2\times 1–5\\ & \Rightarrow & F\left(4\right)=2–5\\ & \Rightarrow & F\left(4\right)=-3\end{array}$

Step4. Finding value for $F\left(5\right)$

Put $x=3in\left(i\right)$

$$\begin{gathered} F(3+2)=2F\left(3\right)-F(3+1) \\ \Rightarrow F\left(5\right)=2F\left(3\right)-F\left(4\right) \\ \Rightarrow F\left(5\right)=2\times 5-(-3) \\ \Rightarrow F\left(5\right)=10+3 \\ \Rightarrow F\left(5\right)=13 \\ \Rightarrow F\left(5\right)=13 \end{gathered}$$

Therefore $F\left(5\right)=13$

Hence, the correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}\mathbf{.}$