Problem 21 - 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. $$ 1.5,-3,6,-12, \dots $$

Step-by-Step Solution

Answer

The general term of the sequence is \(a_{n} = 1.5 * (-2)^{n-1}\) and the seventh term of the sequence is \(a_{7} = -192\).

Step 1: Determine the Common Ratio

Identify the common ratio \(r\) between terms in the sequence by dividing a term in the sequence by its preceding term. Using the first two terms gives \(-3 / 1.5 = -2. The common ratio is \(r=-2\).

Step 2: Write the General Form

The general formula for the nth term of a geometric sequence is \(a_{n}= a_{1}* r^{n-1}\). For this sequence where \(a_{1}=1.5\) and \(r=-2\), the formula is then \(a_{n} = 1.5 * (-2)^{n-1}\).

Step 3: Find the Seventh Term

Use the general formula from Step 2 to find the seventh term by plugging in \(n=7\). The calculation is \(a_{7} = 1.5 * (-2)^{7-1} = -192\)

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key concept that helps determine the relationship between each term in the sequence. It is a constant factor by which each term of the sequence is multiplied to obtain the next term. Understanding how to find the common ratio is essential.

To determine the common ratio, you simply divide any term in the sequence by the previous term. For example, in the sequence 1.5, -3, 6, -12, we find the ratio like this:

• Take the second term \( (-3) \)
• Divide it by the first term \( (1.5) \).
• This results in a common ratio of \( -2 \)

By showing consistency of this ratio between all terms, you confirm it as the common ratio for the sequence. Recognizing this ratio is crucial as it will be used to find subsequent terms in the sequence.

General Term

The formula for the general term, also known as the nth term, of a geometric sequence is essential to understanding how the sequence is constructed. The formula for finding this term is:\[a_n = a_1 \, r^{n-1}\]

Here's the breakdown:

• \( a_n \) represents the nth term you're looking for.
• \( a_1 \) is the first term of the sequence.
• \( r \) stands for the common ratio.
• \( n \) is the term number you're trying to find the value of.

In our example, where \( a_1 = 1.5 \) and \( r = -2 \), the general formula becomes:\[a_n = 1.5 \, (-2)^{n-1}\]This formula allows you to calculate any term in the sequence, as long as you know its position.

Nth Term

The nth term in a geometric sequence is a particular term that can be quickly calculated using the general formula devised earlier. Let’s explore how you would compute a specific term, say the 7th term, from the sequence.

In our example given the formula:\[a_n = 1.5 \, (-2)^{n-1}\]

We plug in \( n = 7 \) to find:

• \(a_7 = 1.5 \, (-2)^{7-1}\)
• Calculate \((-2)^{6} = 64\)
• Then multiply \(1.5 \, \times \, 64\)
• Which results in \(-192\)

This means that the seventh term of the sequence is \(-192\). Being able to calculate the nth term allows you to identify any point within the sequence without needing to list them all.