3.
\begin{figure}[h]
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\caption{Figure 1}
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There are three lily pads on a pond. A frog hops repeatedly from one lily pad to another.
The frog starts on lily pad A, as shown in Figure 1.
In a model, the frog hops from its position on one lily pad to either of the other two lily pads with equal probability.
Let \(p _ { n }\) be the probability that the frog is on lily pad A after \(n\) hops.
- Explain, with reference to the model, why \(p _ { 1 } = 0\)
The probability \(p _ { n }\) satisfies the recurrence relation
$$p _ { n + 1 } = \frac { 1 } { 2 } \left( 1 - p _ { n } \right) \quad n \geqslant 1 \quad \text { where } p _ { 1 } = 0$$
- Prove by induction that, for \(n \geqslant 1\)
$$p _ { n } = \frac { 2 } { 3 } \left( - \frac { 1 } { 2 } \right) ^ { n } + \frac { 1 } { 3 }$$
- Use the result in part (b) to explain why, in the long term, the probability that the frog is on lily pad A is \(\frac { 1 } { 3 }\)