Chapter 1.1, Problem 56E

To determine

To find: the value of $P\left(6\right)$ , if for function $P\left(x\right)$ the value of $P\left(1\right)=1,P\left(2\right)=2,P\left(3\right)=3,$ and $P\left(x+1\right)=\frac{P\left(x-2\right)P\left(x-1\right)+1}{P\left(x\right)}$ , when $x\ge 3$ .

Answer to Problem 56E

  $\frac{4}{7}$ .

Explanation of Solution

Given information:

  $\begin{array}{c}P\left(1\right)=1\mathrm{....}\left(1\right)\\ P\left(2\right)=2\mathrm{....}\left(2\right)\\ P\left(3\right)=3\mathrm{....}\left(3\right)\mathrm{....}\left(1\right)\\ P\left(x+1\right)=\frac{P\left(x-2\right)P\left(x-1\right)+1}{P\left(x\right)}\mathrm{....}\left(4\right)\end{array}$

Calculation:

Compare $P\left(6\right)\text{ and }P\left(x+1\right)$ as follows:

  $\begin{array}{c}x+1=6\\ x=5\end{array}$

Substitute value of x in equation (4) and evaluate $P\left(6\right)$ as shown below.

  $\begin{array}{c}P\left(x+1\right)=\frac{P\left(x-2\right)P\left(x-1\right)+1}{P\left(x\right)}\\ P\left(5+1\right)=\frac{P\left(5-2\right)P\left(5-1\right)+1}{P\left(5\right)}\\ P\left(6\right)=\frac{P\left(3\right)P\left(4\right)+1}{P\left(5\right)}\end{array}$

Using equation (3), the above equation can be rewritten as follows:

  $P\left(6\right)=\frac{3P\left(4\right)+1}{P\left(5\right)}$ … (5)

To solve the above equation, find the values of $P\left(4\right)$ and

  $P\left(5\right)$ as shown below.

Compare $P\left(4\right)\text{ and }P\left(x+1\right)$ as follows:

  $\begin{array}{c}x+1=4\\ x=3\end{array}$

Substitute value of x in equation (4) and evaluate $P\left(4\right)$ as shown below.

  $\begin{array}{c}P\left(3+1\right)=\frac{P\left(3-2\right)P\left(3-1\right)+1}{P\left(3\right)}\\ P\left(4\right)=\frac{P\left(1\right)P\left(2\right)+1}{P\left(3\right)}\end{array}$

Using equations (1),(2), and (3) the above equation can be rewritten as follows:

  $\begin{array}{c}P\left(4\right)=\frac{\left(1\right)\left(2\right)+1}{\left(3\right)}\\ \text{=}\frac{\text{3+3}}{3}\\ =1\end{array}$

Thus, the value of $P\left(4\right)$ is 1.

Similarly, compare $P\left(5\right)\text{ and }P\left(x+1\right)$ as follows:

  $\begin{array}{c}x+1=5\\ x=4\end{array}$

Substitute value of x in equation (4) and evaluate $P\left(5\right)$ as shown below.

  $\begin{array}{c}P\left(4+1\right)=\frac{P\left(4-2\right)P\left(4-1\right)+1}{P\left(4\right)}\\ P\left(5\right)=\frac{P\left(2\right)P\left(3\right)+1}{P\left(4\right)}\end{array}$

Substituting the values of $P\left(2\right),P\left(3\right)$ and $P\left(4\right)$ in the above equation, we get

  $\begin{array}{c}P\left(5\right)=\frac{\left(2\right)\left(3\right)+1}{1}\\ \text{=7}\end{array}$

Thus, the value of $P\left(5\right)$ is 7.

Now put the values of $P\left(4\right)$ and $P\left(5\right)$ in equation (4) to get,

  $\begin{array}{c}P\left(6\right)=\frac{3P\left(4\right)+1}{P\left(5\right)}\\ P\left(6\right)=\frac{3\left(1\right)+1}{7}\\ =\frac{4}{7}\end{array}$

Thus the value of $P\left(6\right)$ is $\frac{4}{7}$ .