8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence.
Given that sales total \(\pounds 8100\) during the first six months, use the model to
- find the value of \(x\),
- find the expected value of sales in the eighth month,
- show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
- Explain why this model cannot be valid over a long period of time.