To determine Mapiya's cumulative earnings [tex]\(E(n)\)[/tex] in dollars after writing [tex]\(n\)[/tex] sequels, follow these steps:
1. Understand the Earnings Pattern:
- For the first book, Mapiya earns [tex]\( \$75,000 \)[/tex].
- For each subsequent book, her earnings double. This cumulative doubling effect can be described as a geometric sequence where each term is twice the previous term.
2. Summing the Series:
- Let's break down the cumulative earnings after each book:
- After the 0th book (first book): [tex]\( \$75,000 \)[/tex]
- After the 1st sequel: [tex]\( \$75,000 + 2 \times 75,000 \)[/tex]
- After the 2nd sequel: [tex]\( \$75,000 + 2 \times 75,000 + 4 \times 75,000 \)[/tex]
- So, for [tex]\(n\)[/tex] sequels, we have a series with each term being a power of 2 starting from 1.
3. Formula for the Geometric Series:
- The earnings follow a geometric series where:
[tex]\[
E(n) = 75,000(1 + 2 + 4 + 8 + \cdots + 2^n)
\][/tex]
- This series can be summed using the formula for the sum of a geometric series:
[tex]\[
S = a \frac{(r^{n+1} - 1)}{r - 1}
\][/tex]
Here, [tex]\(a = 1\)[/tex] (the first term), [tex]\(r = 2\)[/tex] (the common ratio), and the number of terms is [tex]\(n + 1\)[/tex].
4. Applying the Formula:
- Substitute the appropriate values into the formula:
[tex]\[
S = \frac{2^{n+1} - 1}{2 - 1} = 2^{n+1} - 1
\][/tex]
- Multiply by 75,000 to get the total earnings:
[tex]\[
E(n) = 75,000 \cdot (2^{n+1} - 1)
\][/tex]
Thus, Mapiya's cumulative earnings [tex]\(E(n)\)[/tex] after writing [tex]\(n\)[/tex] sequels is given by the formula:
[tex]\[
E(n) = 75,000 \cdot (2^{n+1} - 1)
\][/tex]
Let's verify this formula with an example:
- For [tex]\(n = 3\)[/tex]:
[tex]\[
E(3) = 75,000 \cdot (2^{4} - 1) = 75,000 \cdot (16 - 1) = 75,000 \cdot 15 = 1,125,000
\][/tex]
Thus, the function to calculate Mapiya's cumulative earnings [tex]\(E(n)\)[/tex] after writing [tex]\(n\)[/tex] sequels is:
[tex]\[
E(n) = 75,000 \cdot (2^{n+1} - 1)
\][/tex]
