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

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

Since |r| > 1,

S_n = a ( rⁿ - 1 ) / ( r - 1 )

The sum of the first eight terms is:

S₈ = 11000000 ( 1.55⁸ - 1 ) / ( 1.55 - 1 )

S₈ = 11000000 ( 32316056 ) / ( 0.55 )

S₈ = 646321123

"millions of phones, rounded to one decimal place" is 646.3 million.

Answer 2

EDIT: Wording of the problem made me assume that the rate of sales increased 55% over the span of 7 years. This is not the case since it is provided that the common ratio r was given, making this a much simpler problem... Replace our value of r with 1.55 and follow my same steps outlined below. Thank you David, and onto my next cup of coffee!

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 8th 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 S₈ since 2015 is seven years after 2008, and 2008 is year 1.

Sn = a1(1-rⁿ) / (1 - r)

S₈ = 11,000,000(1-(1.0646)⁸)/(1-(1.0646))

S₈ = 11,000,000(1-1.55)/(1-(1.0646))

S₈ = 11,000,000(-0.55)/(-0.0646)

S₈ = 93,653,250.8

Sn = a1(1-rⁿ) / (1 - r)

S₈ = 11,000,000(1-(1.55)⁸)/(1-(1.55))

S₈ = 11,000,000(-32.316...)/(-.55)

S₈ = 646,320,000

646 Million Phones since 2008.