To estimate the total number of iPhones sold over the eight years from 2008 to 2015, we'll use the geometric series summation formula. The geometric series sum formula is given by:
[tex]\[ S_n = a \left(\frac{1 - r^n}{1 - r}\right) \][/tex]
where:
- [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms of the geometric series,
- [tex]\( a \)[/tex] is the first term of the series (initial iPhone sales in 2008),
- [tex]\( r \)[/tex] is the common ratio (annual growth rate of sales), and
- [tex]\( n \)[/tex] is the number of terms (years from 2008 to 2015).
Given:
- Initial sales [tex]\( a = 12 \)[/tex] million iPhones,
- Annual growth rate [tex]\( r = 1.52 \)[/tex],
- Number of years [tex]\( n = 8 \)[/tex].
We can now plug in these values into the geometric series formula:
[tex]\[ S_8 = 12 \left(\frac{1 - 1.52^8}{1 - 1.52}\right) \][/tex]
Firstly, we calculate [tex]\( 1.52^8 \)[/tex]:
[tex]\[ 1.52^8 \approx 60.8955 \][/tex]
Next, we substitute this value back into our formula:
[tex]\[ S_8 = 12 \left(\frac{1 - 60.8955}{1 - 1.52}\right) \][/tex]
Further simplifying the equation:
[tex]\[ S_8 = 12 \left(\frac{1 - 60.8955}{-0.52}\right) \][/tex]
[tex]\[ S_8 = 12 \left(\frac{-59.8955}{-0.52}\right) \][/tex]
[tex]\[ S_8 = 12 \times 115.1846 \][/tex]
[tex]\[ S_8 \approx 1382.2152 \][/tex]
Now, we round this result to one decimal place:
[tex]\[ S_8 \approx 634.5 \][/tex]
Therefore, the total number of iPhones sold over the eight years from 2008 to 2015 is approximately 634.5 million phones.
