Problem 66 - 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 thirtieth day of the month?

Step-by-Step Solution

Answer

The amount set aside for savings on the thirtieth day of the month is \(\$ 2^{29}\) dollars

Step 1: Identify the characteristics of the geometric sequence.

The first term (a) of the sequence is \(\$1\) and the ratio (r), which is the factor by which we multiply each term to get the next, is \(2\). We are asked to find the thirtieth term of the sequence, which means n (the term number in the sequence) is \(30\).

Step 2: Apply the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is given by \(a_n = a \cdot r^{(n-1)}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Therefore, substitute \(a = 1\), \(r = 2\), and \(n = 30\) into the formula to find the thirtieth term.

Step 3: Compute the value.

Calculating it, we have \(a_{30} = 1 \cdot 2^{(30-1)} = 1 \cdot 2^{29}\). The result is the equivalent in dollars of the value of \(2^{29}\).

Key Concepts

nth term formula

In a geometric sequence, each term is scaled by a constant value known as the common ratio. The formula used to calculate any term in this type of sequence is pivotal for understanding its behavior and predicting future terms. This formula is known as the nth term formula and is expressed as:

• \(a_n = a \cdot r^{(n-1)}\)

Here, \(a\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term number we wish to find. Consider it as a map that guides you from the first day of saving \(\$1\) to projecting how much you will save on any future day by multiplying the common ratio repeatedly.

common ratio

The common ratio in a geometric sequence is the factor by which you multiply each term to get to the next. In our savings problem, the common ratio is \(2\). Each day you save twice as much money as you saved the previous day. This consistency in multiplication is what defines a geometric sequence, distinguishing it from arithmetic sequences where you add a constant value instead.Understanding the common ratio:

• Increases rapidly: Multiplying by a number greater than one will rapidly increase the next terms.
• In this case, the savings plan grows exponentially as days go by.

Grasping this concept is key to predicting future values and understanding the nature of problems involving geometric progressions.

geometric sequence problems

When dealing with geometric sequence problems, one must focus on the key elements like identifying the first term and the common ratio. Knowing these elements allows you to use the nth term formula effectively. In the issue discussed, you start with a small daily saving that grows exponentially throughout the month:

• First term \(a = 1\)
• Common ratio \(r = 2\)

Problems of this nature often ask for predictions of future entries in the sequence or simply the accumulation of these savings over time.Break the problem into manageable steps:

• Identify known values: first term and common ratio.
• Apply the nth term formula to predict the outcome for desired terms.
• Solve to find actual values.

term calculation

Calculating a specific term in a geometric sequence involves a clear understanding of the nth term formula. Let's dive into the term calculation itself:In our savings scenario, to find out how much you will save on the 30th day, you use the nth term formula:

• Given \(a = 1\), \(r = 2\), and \(n = 30\), plug these into the formula: \(a_{30} = 1 \times 2^{29}\)

Perform the computation:

• Calculate \(2^{29}\)
• The result is the amount saved on the 30th day.

By performing these calculations, the predicted savings spell out the power of geometric sequences in practical scenarios, especially those involving exponential growth.