What is the long-term share of the detergent market achieved by the brand ‘Number One’, described in Question 5.

Data from Question 5

A detergent manufacturer wishes to forecast its future sales. The market research department assess that their ‘Number One’ brand has 20% of the potential market at present. They also estimate that 15% of those who bought ‘Number One’ in a given month will buy a different detergent in the following month and that 35% of those who bought a rival brand will buy ‘Number One’ in the next month. Show that their share P_n% of the market in the nth month satisfies the recurrence relation

Find the values of P_n for n = 1, 2, 3 and 4 and illustrate them on an appropriate diagram.

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Pn+1= 35 +0.5P, with Po = 20