A company is considering two perpetual advertising contracts
MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series
Problem
A company is considering two perpetual advertising contracts. Payments are made once per year, beginning one year from today, and continue indefinitely.
Contract A requires a first-year payment of $10,000. Each subsequent yearly payment is 10% smaller than the previous payment.
Contract B requires a first-year payment of $6,000. Each subsequent yearly payment is 5% larger than the previous payment.
Assume the company uses a constant discount rate of 8% per year when computing present value. The present value of a contract equals the sum of the discounted future payments.
(1) Write an explicit infinite series representing the present value of Contract A. Clearly identify the first term and common ratio of the series you construct.
(2) Write an explicit infinite series representing the present value of Contract B in analogous fashion.
(3) Determine whether each of the series from parts (a) and (b) converges. Justify your conclusions by referencing necessary conditions for convergence.
(4) Compute the present value of each contract to the nearest whole dollar.
(5) Based on your calculations, recommend which contract the company should choose to minimize total cost, and explain your reasoning in context.