Problem 14 - Algebra for College Students

Question

Eric saved a nickel the first day of a month, a dime the second day, and 20 cents the third day and then continued to double his daily savings each day for 14 days. What were his daily savings on the 14th day? What were his total savings for the 14 days? \(\quad \$ 409.60 ; \$ 819.15\)

Step-by-Step Solution

Answer

On the 14th day, Eric saved $409.60. His total savings for the 14 days was $819.15.

Step 1: Depict the Sequence of Savings

Eric saves money in the form of a sequence where the saving doubles each day starting from the third day. The first day he saves 5 cents, represented by \(0.05\) USD. Similarly, the second day he saves a dime, represented by \(0.10\) USD. The third day he starts doubling the savings of the previous day, which is 20 cents or \(0.20\) USD.

Step 2: Calculate Savings for Each Day

To find the savings on each day after the third day, you double the previous day's savings. The sequence of savings in dollars is: \(0.05, 0.10, 0.20, 0.40, 0.80, 1.60, 3.20, 6.40, 12.80, 25.60, 51.20, 102.40, 204.80, 409.60\). Each day is double the savings of the previous day starting from 20 cents (0.20 USD).

Step 3: Determine Daily Savings on the 14th Day

The formula for the nth term of this geometric sequence can be used: \(a_n = a_1 \cdot 2^{(n-1)}\), where \(a_1 = 0.20\) USD (the third day savings as the base of doubling). For the 14th day, \(a_{14} = 0.20 \times 2^{(14-3)} = 409.60\) USD.

Step 4: Calculate Total Savings for 14 Days

To find the total savings, sum all terms from the first to the 14th day: \( 0.05 + 0.10 + 0.20 + 0.40 + 0.80 + 1.60 + 3.20 + 6.40 + 12.80 + 25.60 + 51.20 + 102.40 + 204.80 + 409.60 = 819.15 \) USD.

Key Concepts

Doubling Sequence

A doubling sequence is a type of geometric progression where each term in the sequence is twice the previous one. This sequence is quite simple to understand and calculate because you merely multiply the preceding number by two to get the next number. In the context of Eric's savings, starting from the third day onwards, his daily savings form a doubling sequence.

• On Day 3: The sequence begins with 0.20 USD.
• On Day 4: The savings double, making it 0.40 USD.
• On Day 5: Again doubling to 0.80 USD.

This pattern continues, meaning each subsequent day's savings are just a simple multiplication of the previous day's savings by 2. Doubling sequences are powerful in depicting exponential growth, commonly seen in cases like this savings problem, bacteria reproduction, and technology improvements.

Geometric Progression

A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For Eric's savings, this ratio is 2. The formula used to find the nth term of a geometric sequence simplifies the process of finding any term in the sequence:\[ a_n = a_1 \cdot r^{(n-1)} \]where:

• \(a_n\) is the nth term
• \(a_1\) is the first term of the geometric progression
• \(r\) is the common ratio
• \(n\) is the term number

In Eric's case, if you consider starting from the third day with 0.20 USD, the calculation on the 14th day would be:\[ a_{14} = 0.20 \times 2^{(14-3)} = 409.60 \] USD.Geometric progressions are extensive in various fields like finance, computer science, and physics due to their straightforward yet powerful portrayal of exponential change.

Savings Problem

Problems involving savings often showcase practical applications of mathematical concepts like geometric progressions and doubling sequences. The key theme in a savings problem is tracking accumulations over time, highlighting both growth and rate of increase.In Eric's savings:

• Daily savings are increasing exponentially, teaching the essence of compounding and exponential growth.
• The total sum, calculated by adding each day's savings, reflects the importance of controlled long-term planning—a typical real-world financial consideration.

The total savings gained over 14 days is obtained by adding all the individual daily savings. The formula for the sum of a geometric series can determine this total quickly:\[ S_n = a_1 \frac{1-r^n}{1-r} \]for significant computation where

• \(S_n\) is the sum of the first n terms
• \(r\) is the common ratio (2, in this case)

By manually adding each day’s savings totaling 819.15 USD, it illustrates the practicality and necessity of knowing how series work in real-world financial problems, similar to practices like setting aside savings and understanding compound interest.