Question

In Exercises 15-22, write a rule for the \(n\)th term of the sequence. Then find \(a_7\). \(375,75,15,3, \ldots\)

Short answer

The 7th term of the sequence, \(a_7\), is 0.024.

Step-by-step solution

Identify the common ratio

In order to determine whether this sequence is indeed a geometric sequence, one needs to identify if there is a constant ratio between subsequent terms. By dividing each term by its preceding term, it can be seen that this ratio is 5. So, the common ratio \(r = 5\).

Define the general rule of the geometric sequence

The general rule or formula for any term in a geometric sequence is \(a_n = a_1*(r^{(n-1)})\), where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(r\) is the common ratio and \(n\) is the term position. For this exercise, the first term \(a_1 = 375\) and the common ratio \(r = 5\). So, the rule for the \(n\)th term of the sequence is \(a_n = 375*(5^{-(n-1)})\).

Calculate \(a_7\)

Substituting the values of \(n = 7\) into the formula, we get: \(a_7 = 375*(5^{-(7-1)}) = 375*(5^{-6})\). Evaluating this expression gives \(a_7 = 375*(\frac{1}{5^6}) = 375*(\frac{1}{15625}) = 0.024\).

Key concepts

Common Ratio

The common ratio is a fundamental element in understanding geometric sequences. In essence, it is the factor by which we multiply one term of the sequence to get the next term. To find the common ratio, divide each term by the term that precedes it. If you consistently get the same number, then this is your common ratio, and the sequence is indeed geometric. For example, in the sequence provided, which begins with 375, 75, 15, 3, …, dividing 75 by 375 gives \[ \frac{75}{375} = \frac{1}{5} \]. Similarly, dividing 15 by 75 also gives \(\frac{1}{5}\), and dividing 3 by 15 provides the same result. This consistent result tells us that the common ratio \(r\) is \(\frac{1}{5}\). Common ratios can be any non-zero number:

• If it is greater than 1, terms will grow larger.
• If it is between 0 and 1, terms will shrink.
• Negative ratios will alternate terms between positive and negative values.

General Term Formula

The general term formula for a geometric sequence is a useful tool to find any term in the sequence without listing all the terms. This formula is expressed as \[ a_n = a_1 \times (r^{n-1}) \], where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence.This formula allows us to find out any term directly. If we know \(a_1\) and \(r\), we can plug in any value of \(n\) to find the corresponding term.Consider the sequence from the exercise:

• First term \(a_1 = 375\)
• Common ratio \(r = \frac{1}{5}\)

Substitute these into the formula: \[ a_n = 375 \times (\frac{1}{5})^{(n-1)} \].Now, you can calculate any term of the sequence by simply substituting the desired \(n\) value into this formula.

Sequence Rule Derivation

Deriving the sequence rule involves understanding how the formula for the general term is constructed and applying it to specific terms in a given sequence. To derive the sequence rule: start by identifying the first term of the sequence, which in our example, is 375. Then, confirm it has a constant common ratio by dividing each term by the previous term; this ratio was confirmed as \(\frac{1}{5}\). Put it into the standard form of the geometric sequence expression:

• \(a_1 = 375\) (the first term)
• \(r = \frac{1}{5}\) (the common ratio)

The rule for any term in the sequence is then: \[ a_n = 375 \times (\frac{1}{5})^{n-1} \].To find the seventh term, substitute \(n = 7\) into the sequence rule:\[ a_7 = 375 \times (\frac{1}{5})^{7-1} = 375 \times (\frac{1}{5})^6 \].Calculating this gives us the seventh term. Sequence rule derivation helps consolidate your understanding of sequence, as it involves identifying pattern components and applying them mathematically.