Problem 65 - Algebra and Trigonometry

Question

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) 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

You will put aside \$16384 on the fifteenth day of the month.

Step 1: Identify the first term, common ratio, and term number

The first term 'a' is the amount saved on the first day, which is \$1. The common ratio 'r' is how much the savings increase each day, which is stated to be double, or 2. The term number 'n' is the day number we want to find the savings for, that is day 15.

Step 2: Use the geometric sequence formula

We put the values we identified into the formula for the nth term of a geometric sequence, which gives us the formula \(1 * 2^{(15-1)}\)

Step 3: Calculate the savings using the formula

We calculate the expression \(1 * 2^{(15-1)}\) which yields 16384.

Key Concepts

General Term

In a geometric sequence, the general term is used to find any term in the sequence without needing to list all preceding terms. It helps identify how each term in the sequence is related systematically to the previous terms. The general term is a powerful tool because once you know it, you can calculate any term of the sequence quickly and efficiently. In this problem's context, the general term allows you to calculate how much you will save on any specific day, like the fifteenth. To establish this, it is crucial to first completely understand the rule determining the sequence's progression.
The general term formula for a geometric sequence is found by applying the nth term formula, which serves as the framework for understanding the sequence's progression.

Nth Term Formula

The nth term formula for a geometric sequence is a mathematical expression used to determine the value of any term in the sequence based on its position. The formula is derived as follows:

• The first term of the sequence, represented by \(a\)
• The common ratio between consecutive terms, represented by \(r\)
• The term number you are going to find, represented by \(n\)

The formula is expressed as:\[ a_{n} = a imes r^{(n-1)} \]
This formula simplifies the process of finding any term in a geometric sequence by providing a straightforward calculation involving these fundamental components. It allows you to find the savings on day 15 by plugging the values into this formula. This leads us into calculating how much will be saved specifically on the fifteenth day with the given sequence.

Common Ratio

The common ratio in a geometric sequence is a constant factor that each term has multiplied to get the next term. In this problem, the savings amount doubled each day. This doubling pattern indicates that the common ratio is 2.
To identify the common ratio, compare two consecutive terms in the sequence, such as day two savings divided by day one savings. The formula to compute the common ratio \(r\) is:\[ r = \frac{a_{n}}{a_{n-1}} \]
For this specific example, if you start with \(1\), and the savings doubles the next day to \(2\), calculating \(\frac{2}{1}=2\) confirms the common ratio. Knowing this ratio is essential for formulating the nth term and thereby predicting future terms in the sequence.

First Term

The first term in a geometric sequence is the anchor point from which the rest of the sequence builds. It sets the starting amount for how often and by how much it needs multiplying to form the subsequent terms. For this exercise, the first term is the initial amount saved on the first day, which is \(1\).
This first term is articulated in the formula as the base point \(a\). It is crucial in calculating any other future terms because every calculation builds from this initial value. Beginning your understanding with this starting point is essential for using the nth term formula accurately and deriving the correct sequence flow.
Understanding the first term's role sharply can help you see how each day simply builds exponentially upon the previous, starting from day one.