- In a model for the number of subscribers to a new social media channel it is assumed that
- each week \(20 \%\) of the subscribers at the start of the week cancel their subscriptions
- between the start and end of week \(n\) the channel gains \(20 n\) new subscribers
Given that at the end of week 1 there were 25 subscribers,
- explain why the number of subscribers at the end of week \(n , U _ { n }\), is modelled by the recurrence relation
$$U _ { 1 } = 25 \quad U _ { n + 1 } = 0.8 U _ { n } + 20 ( n + 1 ) \quad n = 1,2,3 , \ldots$$
- Prove by induction that for \(n \geqslant 1\)
$$U _ { n } = 325 \left( \frac { 4 } { 5 } \right) ^ { n - 1 } + 100 n - 400$$
Given that 6 months after starting the channel there were approximately 1800 subscribers,
- evaluate the model in the light of this information.