Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Term of a Geometric Sequence Find a formula for the \(n\)th term of the geometric sequence. Then find the indicated term of the geometric sequence. 9th term: \(7,21,63, \dots\)

Step-by-Step Solution

Answer

The \(n\)th term of the given geometric sequence is \(7 \cdot 3^{(n-1)}\). The 9th term of the sequence is \(7 \cdot 3^{8}\), or 150807.

Step 1: Understand the Formula for 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. The formula for the \(n\)th term of a geometric sequence is \(a \cdot r^{(n-1)}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Identify the First Term and Common Ratio

In the given sequence, the first term \(a\) is 7. The common ratio \(r) can be found by dividing any term by the previous term. For instance, \(21 ÷ 7 = 3\) or \(63 ÷ 21 = 3\). Therefore, the common ratio \(r\) is 3.

Step 3: Substitute Into the Formula

Substitute the values \(a = 7\), \(r = 3\) into the geometric sequence formula to find the formula for the \(n\)th term, which becomes \(7 \cdot 3^{(n-1)}\).

Step 4: Find the 9th Term

Substitute \(n = 9\) in the formula to find the 9th term. So, the 9th term will be \(7 \cdot 3^{(9-1)} = 7 \cdot 3^{8}\).

Key Concepts

Common Ratio

In a geometric sequence, the magic number that ties the sequence together is known as the 'common ratio'. This ratio is the factor that each term is multiplied by to get the next term in the sequence. To identify the common ratio, simply divide any term in the sequence by the previous term.

For example, in the sequence you are working with \(7, 21, 63, \ldots\), the common ratio can be calculated by taking \(21 \div 7 = 3\) and similarly, \(63 \div 21 = 3\). Hence, the common ratio is 3.

The common ratio essentially dictates the pace at which the sequence grows or shrinks, ensuring a constant multiplicative relationship among terms.

• It must be a non-zero number to ensure that the sequence is meaningful.
• It defines whether the sequence is increasing (when \(r > 1\)) or decreasing (when \(r < 1\)).
• If \(r = 1\), the sequence is constant.

Nth Term Formula

The formula for finding the nth term in a geometric sequence is your key to unlocking any term in the sequence. It's expressed as: \(a \cdot\ r^{(n-1)}\) where \(a\) represents the first term, \(r\) is the common ratio, and \(n\) is the term number you want to find. This formula enables you to find any term in the sequence without listing all terms.

Here's how each part functions:

• \(a\): The starting point of your sequence. In your example, \(a = 7\).
• \(r\): Describes the repetitive multiplicative factor between terms. Here, \(r = 3\).
• \(n-1\): The exponent \(n-1\) determines how many times the common ratio is used, starting after the first term. This calculation helps to jump directly to any term in the sequence.

The quick calculation for the nth term keeps your work neat and efficient, especially as sequences become extensive.

In your exercise, substituting \(a = 7\) and \(r = 3\) gives the formula: \(7\cdot3^{(n-1)}\).

Sequence Terms

Sequence terms refer to the individual elements or numbers within a sequence. In a geometric sequence, each term is a result of multiplying the previous term by the common ratio. Understanding sequence terms is essential for identifying patterns and predicting future terms without listing every single one.

For the sequence \(7, 21, 63, \ldots\):

• The first term is 7. This serves as your original starting point.
• Subsequent terms are generated by multiplying the previous term by the common ratio, which is 3 in this instance.
• Every term in the sequence can be predicted using the nth term formula rather than manually multiplying each time.

To find any specific term, such as the 9th term, plug in the term number into the nth term formula \(7 \cdot\ 3^{(n-1)}\). Applying this for the 9th term, the calculation is \(7 \cdot\ 3^{8}\). This method ensures an accurate and efficient approach to accessing any term, especially when dealing with large sequences.