Problem 57 - College Algebra

Question

Use the formula for the general term (the nth term) of a geometric sequence to solve. Isuppose you Save 1 dollar the first day of a month, 2 dollar the second day, 4 dollar the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the fifteenth day of the month?

Step-by-Step Solution

Answer

On the fifteenth day of the month, you will put aside 16384 dollars for savings.

Step 1: Identify the Type of Sequence

In this problem, the amount of money saved each day is doubling, which is a classic example of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is 2.

Step 2: Apply the nth Term Formula

The formula for the nth term of a geometric sequence is \(a_n = a_1 * r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Here, \(a_1 = 1\), \(r = 2\), and \(n = 15\). Substituting these values into the formula gives us \(a_{15} = 1 * 2^{(15-1)}\). Simplifying that, we get \(a_{15} = 2^{14}\).

Step 3: Calculate the Amount Saved

Now, calculate \(2^{14}\) to find the amount saved on the fifteenth day. The result is 16384 dollars.

Key Concepts

nth term formula

In the realm of geometric sequences, understanding how to determine the value of a specific term is crucial. This is where the **nth term formula** shines. The nth term formula provides a way to calculate any term in a sequence without listing all previous terms. The formula is given by \[ a_n = a_1 \times r^{(n-1)} \]where:

• \( a_n \) is the term we wish to find,
• \( a_1 \) represents the first term of the sequence,
• \( r \) is the **common ratio**, or the factor by which each term is multiplied to obtain the subsequent term,
• \( n \) is the position of the term in the sequence.

Applying this formula is straightforward. Simply plug in the known values for the first term, the common ratio, and the position to calculate the desired term. For instance, to find the amount saved on the fifteenth day, use \( a_1 = 1 \), \( r = 2 \), and \( n = 15 \) to compute \( a_{15} = 1 \times 2^{14} \). This results in aiding in quickly finding the 15th term without calculating each term individually.

common ratio

The **common ratio** is a defining feature of a geometric sequence. It's the constant factor between consecutive terms. In simpler words, each term is obtained by multiplying the previous term by a fixed number, known as the common ratio, denoted by \( r \).In our example of saving money, you start with \\(1 on the first day and double it every subsequent day. Thus, the common ratio is \( 2 \). Each day, multiply the savings of the previous day by two to find the savings for the current day. This is what defines the sequence as geometric:

• Day 1: \\)1
• Day 2: \\(2 (1 \times 2)
• Day 3: \\)4 (2 \times 2)
• Day 4: \$8 (4 \times 2)

The basics of a geometric sequence rely heavily on understanding this ratio. Whether it's doubling, tripling, or any other multiplication factor, the common ratio is your guide to finding your next term.

sequence pattern observation

Observing the pattern in a sequence can offer a powerful way to understand and predict the behavior of a sequence. In a geometric sequence, the pattern is straightforward due to its reliance on a common ratio.By closely examining the terms of the sequence, it becomes evident how each term relates to its predecessor. Consider an example where each day's saving doubles compared to the previous day. This doubling pattern is consistently observed:

• From \\(1 to \\)2, the increase is by multiplying \( 1 \times 2 \).
• From \\(2 to \\)4, again, it's \( 2 \times 2 \).
• Continuing with \\(4 to \\)8, it's consistent with \( 4 \times 2 \).

Recognizing and understanding these patterns allow for faster identification of subsequent terms, without the monotonous task of manual multiplication. The visual sequence of terms itself serves as evidence of the property's correctness, supporting both intuitive and formal comprehension.