Problem 50 - Algebra for College Students

Question

If you deposit 1 cent into your piggy bank on September 1 and each day thereafter deposit twice as much as on the previous day, then how much will you be depositing on September \(30 ?\) The total amount deposited for the month can be found without adding up all 30 deposits. Look at how the amount on deposit is increasing each day and see whether you can find the total for the month. Give reasons for your answers.

Step-by-Step Solution

Answer

On September 30, the deposit is \(5368709.12. The total amount deposited in the month is \)10737418.23.

Step 1: Determine the Daily Deposit Pattern

The daily deposit amount forms a geometric sequence where each day the deposit is twice the amount of the previous day. Let the deposit on the first day, September 1, be denoted by the term: \( a_1 = 0.01 \) dollars (1 cent).

Step 2: Express the Deposit on Day n

The deposit on any subsequent day, n, can be represented as: \( a_n = a_1 \times 2^{(n-1)} \).

Step 3: Calculate the Deposit on September 30

To find the deposit on September 30 (n = 30), substitute 30 into the formula: \( a_{30} = 0.01 \times 2^{29} \). Compute \(2^{29}\) which gives: \( a_{30} = 0.01 \times 536870912 = 5368709.12\) dollars.

Step 4: Sum of the Deposits for the Month

The sum of the deposits forms a geometric series. The formula for the sum \( S_n \) of the first n terms of a geometric series is: \( S_n = a_1 \frac{1 - r^n}{1 - r} \), where \(r\) is the common ratio. Here, \( a_1 = 0.01 \) and \( r = 2 \), and n = 30, so: \[ S_{30} = 0.01 \frac{1 - 2^{30}}{1 - 2} = 0.01 \times (2^{30}-1) = 0.01 \times (1073741824 - 1) = 0.01 \times 1073741823 = 10737418.23 \] dollars.

Key Concepts

geometric series

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our case, we are depositing twice the amount each day compared to the previous day. This doubling pattern represents a common ratio of 2. Starting with a small initial deposit of 1 cent on the first day, we follow this pattern to find the amount for subsequent days.
For instance:

• Day 1: 1 cent
• Day 2: 2 cents
• Day 3: 4 cents
• Day 4: 8 cents
• and so on…

This geometric progression continues, and we can use the formula to easily calculate any term within the series. Knowing how geometric sequences work is essential for understanding the overall deposit pattern and calculating large values efficiently without manually adding each day's deposit.

daily deposit pattern

The daily deposit pattern we are discussing is based on a geometric sequence. Each day, you deposit an amount that is twice as much as the day before. The initial deposit, denoted as the first term (\(a_1\)), is 1 cent, which translates to \(0.01\) dollars. The amount deposited on any given day, \(n\), can be expressed as:
\[ a_n = 0.01 \times 2^{(n-1)} \] Here's a quick breakdown:

• For Day 1, \(a_1 = 0.01\) (1 cent)
• For Day 2, \(a_2 = 0.01 \times 2^1 = 0.02 \) (2 cents)
• For Day 3, \(a_3 = 0.01 \times 2^2 = 0.04 \) (4 cents)
• …
• For Day 30, \(a_{30} = 0.01 \times 2^{29}\)

This pattern increases rapidly due to the exponential growth characteristic of geometric sequences. By plugging in the values, you can quickly determine the amount deposited on any particular day rather than calculating each step manually.

sum of series

When dealing with geometric series, you can also sum all terms to find the total of the series without adding each term separately. The sum of the first \(n\) terms of a geometric series can be found using the formula:
\[ S_n = a_1 \frac{1 - r^n}{1 - r} \] In our case:

• \(a_1 = 0.01\)
• \(r = 2\)
• \(n = 30\)

Substituting these values, we get:
\[ S_{30} = 0.01 \frac{1 - 2^{30}}{1 - 2} = 0.01 \times (2^{30} - 1) \] Compute \(2^{30}\) which is \(1073741824\). Thus:
\[ S_{30} = 0.01 \times (1073741824 - 1) = 0.01 \times 1073741823 = 10737418.23 \] dollars.
This formula helps us quickly find the total amount deposited over a month without laboriously adding each day's deposit individually. Understanding this concept of adding up a geometric series is crucial for efficiently solving problems involving exponential growth, such as our daily deposit pattern.