Johb has purchased 20 books the 1st book is costs 1 dollor the 2nd, 2 dollars the 3rd 4 dollars and the 4th 8 dollors and so on how much did John pay for the 20 books?

This is a geometric progression problem. The price of each book follows a pattern where the cost doubles for each successive book:

- The 1st book costs \$1.
- The 2nd book costs \$2.
- The 3rd book costs \$4.
- The 4th book costs \$8.
- And so on.

The price of the books forms a geometric sequence with the first term $ a = 1 $ and the common ratio $ r = 2 $.

The formula for the sum $ S_n $ of the first $ n $ terms of a geometric series is given by:

\[
S_n = \frac{a(r^n - 1)}{r - 1}
\]

where:
- $ a = 1 $ is the first term,
- $ r = 2 $ is the common ratio,
- $ n = 20 $ is the number of books.

Let's calculate how much John paid for the 20 books:

\[
S_{20} = \frac{1(2^{20} - 1)}{2 - 1} = 2^{20} - 1
\]

Now, let's compute $ 2^{20} - 1 $.The total amount John paid for the 20 books is \$1,048,575.

Would you like further details on this calculation, or any other clarifications?

Here are 5 related questions you could explore:
1. How would the total cost change if John bought 30 books instead of 20?
2. What if the price started at \$2 and doubled for each subsequent book?
3. Can you derive the general formula for the sum of prices if the common ratio is different?
4. How does the sum of a geometric series differ from an arithmetic series?
5. What is the significance of the common ratio in a geometric progression?

**Tip:** The sum of a geometric series grows very quickly when the common ratio is greater than 1, due to exponential growth.

Johb has purchased 20 books. The 1st book costs 1 dollar, the 2nd, 2 dollars, the 3rd 4 dollars, and the 4th 8 dollars, and so on. How much did John pay for the 20 books?

Discover how to calculate the total cost John paid for 20 books, where each book's price doubles the previous one, following a geometric progression. This problem covers the concept of exponential growth and the formula for summing a geometric series. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation