Problem 24 - Algebra and Trigonometry

Question

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 0.0007,-0.007,0.07,-0.7, \ldots $$

Step-by-Step Solution

Answer

The formula for the nth term of the sequence is \(a_{n} = 0.0007 \times (-10)^{n-1}\). Using this formula, the seventh term of the sequence is \(-700\).

Step 1: Identify the first term and the common ratio

Observe the sequence: \(0.0007,-0.007,0.07,-0.7, \ldots\). The first term \(a_{1}\) in the sequence is \(0.0007\). To find the common ratio, divide the second term by the first term. This implies that \(r = -0.007 / 0.0007 = -10\).

Step 2: Write a formula for the nth term

Let's write down the formula for the nth term of a geometric sequence. Using the formula \(a_{n} = a_{1} \times r^{n-1}\), and substituting \(a_{1} = 0.0007\) and \(r = -10\), we get: \(a_{n} = 0.0007 \times (-10)^{n-1}\).

Step 3: Use the formula to find the seventh term

We have the formula, we can now find the seventh term in the sequence by substituting \(n = 7\) into the formula: \(a_{7} = 0.0007 \times (-10)^{7-1}\). Simplifying, we get \(a_{7} = -700\).

Key Concepts

Common Ratio

Understanding the common ratio in a geometric sequence is crucial as it is essentially the backbone of the sequence's progression. The common ratio is the constant factor between consecutive terms of a geometric sequence. To find it, divide any term by the preceding one. In our exercise, for instance, the second term is \( -0.007 \) and the first term is \( 0.0007 \). Dividing these, \( r = -0.007 / 0.0007 = -10 \), we determine that the common ratio is \( -10 \), which means each term is \( -10 \) times the previous one. This ratio is applied iteratively to produce each subsequent term in the sequence.

Let’s elaborate on its importance: The common ratio not only helps to determine the pattern of a sequence but also aids in predicting future terms and establishing a formula for any term in the sequence. If the common ratio is positive, the terms will all have the same sign; if it is negative—as in our case—it causes the terms to alternate in sign.

General Term of a Sequence

In sequences, the general term, often denoted as \( a_n \), provides a mathematical way to find any term in the sequence without listing all the terms. For a geometric sequence, the general term formula is \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) denotes the term number. This formula encapsulates the essence of the sequence, enabling you to jump to any term with ease.

With a sequence's first term and its common ratio, you're equipped to generate any term. For instance, in our exercise, we established the general formula as \( a_n = 0.0007 \times (-10)^{(n-1)} \) by plugging in the first term, \( 0.0007 \) and the common ratio, \( -10 \). Applying this formula eliminates the need to calculate each term manually.

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It exhibits exponential growth or decay, which is dependent on the absolute value of the common ratio: if the ratio is greater than one, we see exponential growth; if it’s between zero and one, there’s exponential decay. In cases like our example sequence, \( 0.0007,-0.007,0.07,-0.7, \ldots \), the common ratio causes the terms to increase in magnitude but alternate the sign had because of its negativity.

Nth Term

The nth term of a sequence represents a formula that directly calculates the term at any given position \( n \) in the sequence. In our geometric sequence, obtaining the nth term is crucial for working with series beyond manual enumeration. The nth term formula for the geometric sequence given as \( a_n = a_1 \times r^{(n-1)} \) is powerful because it embodies the entire sequence in a singleexpression. Exprapolating from our previous steps, you can find, for example, the 7th term by substituting \( n \) with 7 in the formula, resulting in \( a_7 = 0.0007 \times (-10)^{6} \) which simplifies to \( a_7 = -700 \). This demonstrates the convenience of the nth term formula when dealing with sequences.