In the year 2008 Apple sold 11 million iPhones. Over the next few years, annual sales increased by an average of 55%

Question

In the year 2008 Apple sold 11 million iPhones. Over the next few years, annual sales increased by an average of 55%, which we will model using a geometric sequence with r = 1.55

Use the geometric summation formula Sn = a1 (1-r^n) / 1 - r, to estimate the total number of iPhones sold in the eight years spanning 2008 and 2015. Give your answer in millions of phones, rounded to one decimal place.

___________ million phones

Answer 1

It is strange to me that the problem describes "the next few years" as a 7 year gap (my guess is that this question is part of a question bank with different values), but using this information we can find the sum of the geometric series.

The sales increased by an average of 55% over the course of 7 years, and this growth was exponential (not linear) since we are dealing with a geometric series. Thus, the rate of sales (per year) can be found by finding the 7th root of 1.55 (1.55 representing 55% growth over 7 years), which is approximately 1.0646. This is our r, or our growth in sales per year. To calculate S_n, we must find S₇ since 2015 is seven years after 2008.

Sn = a1(1-rⁿ) / (1 - r)

S₇ = 11,000,000(1-(1.0646)⁷)/(1-(1.0646))

S₇ = 11,000,000(1-1.55)/(1-(1.0646))

S₇ = 11,000,000(-0.55)/(-0.0646)

S₇ = 93,653,250.8

Rounding to the nearest million phones... 94 million phones.