Question

Problem: A cell phone company plans to market their new phone to the public. They have already sold 310 units during the first week of the campaign. They plan to increase sales by 17% each week. For example, they plan to sell about 363 units during week.

They want to continue this for a year. Is this an Arithmetic or Geometric Series? I hope you agree this is a Geometric series. Write the first term, the common ratio, and an expression for the recursive rule. Use the actual value of the common ratio in the recursive rule, not just “r”. Write the explicit rule so that you can find any term in the series without finding the previous terms first. How many phones will be sold during week number 37? Show all your work. Round your answer to the nearest whole number of cell phones.

The payroll department wants to know how many sales they can expect during the first year of the campaign. Assume there are 52 weeks in a year. How many phone will be sold total during the first year? Write and evaluate the expression for the correct partial sum of the series. Show all your work. Round your answer to the nearest whole number.

Answer 1

The 1^st week’s sales is 310 units and the 2^nd week’s sales is 17 % higher or, 363 units. Since the ratio of any 2 cosecutive terms is constant (1.17), this is a geometric series. The 1^st tem is 310, the common ratio is 1.17. The geometric series is 310, 310*1.17, 310*(1.17)²,….The nth term of this series is 310(1.17)^n-1.

When n = 37, the no. of phones sold will be 310*(1.17)³⁶ = 310*284.8990643 = 88319 ( on rounding off to the nearest whole number).

The vsum of the 1^st 52 terms of the geometric seties is 310*[1-(1.17)⁵²]/(1-1.17) = 310*3511.892103/ 0.17 = 6404039( on rounding off to the nearest whole number). Thus, 6404039 phones will be sold total during the first year.

Note:

The nth term and the sum of 1^st n terms of the geometric series a,ar,ar²,…. are ar^n-1and a(1-rⁿ)/(1-r) respectively ( if r is not equal to 1).