Review question

# Where will this robot be after $n$ iterations? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8203

## Question

A particular robot has three commands:

• $\mathbf{F}$: Move forward a unit distance;
• $\mathbf{L}$: Turn left $90^{\circ}$;
• $\mathbf{R}$: Turn right $90^{\circ}$.

A program is a sequence of commands. We consider particular programs $P_n$ (for $n \ge 0$) in this question. The basic program $P_0$ just instructs the robot to move forward: $P_0 = \mathbf{F}.$

The program $P_{n+1}$ (for $n \ge 0$) involves performing $P_n$, turning left, performing $P_n$ again, then turning right: $P_{n+1}= P_n\, \mathbf{L}\, P_n\, \mathbf{R}.$

So, for example, $P_{1} = \mathbf{F\, L\, F\, R}$.

1. Write down the program $P_2$.

2. How far does the robot travel during the program $P_n$? In other words, how many $\mathbf{F}$ commands does it perform?

3. Let $l_n$ be the total number of commands in $P_n$; so, for example, $l_0 = 1$ and $l_1 = 4$.

Write down an equation relating $l_{n+1}$ to $l_n$. Hence write down a formula for $l_n$ in terms of $n$. No proof is required. Hint: consider $l_n + 2$.

4. The robot starts at the origin, facing along the positive $x$-axis. What direction is the robot facing after performing the program $P_n$?

5. The first diagram below shows the path the robot takes when it performs the program $P_1$. On the second (blank) diagram, draw the path it takes when it performs the program $P_4$.

6. Let $(x_n, y_n)$ be the position of the robot after performing the program $P_n$, so $(x_0, y_0) = (1,0)$ and $(x_1, y_1) = (1,1)$. Give an equation relating $(x_{n+1}, y_{n+1})$ to $(x_n, y_n)$.

What is $(x_8, y_8)$? What is $(x_{8k}, y_{8k})$?