Question

In Exercises 33-40, write a rule for the \(n\)th term of the geometric sequence. $$ a_2=-48, a_5=\frac{3}{4} $$

Short answer

The rule for the nth term of the given geometric sequence is \( a_n = a_1*r^{n-1} \) where \( a_1 = \frac{a_2}{r} \) and \( r = \sqrt[3]{\frac{a_5}{a_2}} \). Replace the values of \( a_2 \) and \( a_5 \) to get specific values for \( a_1 \) and \( r \). You can enter any positive integer for \( n \) in \( a_n \) to get the corresponding term in the sequence.

Step-by-step solution

Calculate the common ratio

To find the common ratio \( r \), divide the fifth term \( a_5 = \frac{3}{4} \) by the second term \( a_2 = -48 \), then take the cube root of the result since \( a_5 \) is three terms after \( a_2 \). This arises from the principle that in a geometric sequence, any term divided by the previous term gives the common ratio. Here, \( r = \sqrt[3]{\frac{a_5}{a_2}} \)

Calculate the first term

To find the first term \( a_1 \), divide the second term \( a_2 = -48 \) by the common ratio \( r \). This occurs because, in a geometric sequence, any term divided by the common ratio gives the previous term. \( a_1 = \frac{a_2}{r} \)

Formulate the rule for the nth term

Finally, to define a rule for the nth term, use the formula for the nth term of a geometric sequence which is \( a_n = a_1*r^{n-1} \). Consider \( a_1 \) and \( r \) as just identified and n as any positive integer. This expression will represent a rule to calculate any term in the geometric sequence.

Key concepts

Common Ratio of Geometric Sequence

The common ratio in a geometric sequence is a consistent factor that determines the scale of progression from one term to the next within the sequence. It is essentially the multiplier applied to a term to obtain the subsequent term. Finding the common ratio is a critical step when dealing with geometric sequences, as it allows you to predict future terms or backtrack to earlier terms.

In practice, to calculate the common ratio (\r), you divide any term in the sequence by the term that immediately precedes it (\r = \frac{a_{n+1}}{a_n}). When given nonconsecutive terms of a sequence, like in the provided exercise, you can still find the common ratio by taking the nth root of the quotient, where n represents the number of terms in between. For example, to find the common ratio given the second and fifth terms, you would take the cube root of their quotient since there are three terms apart (\r = \(\sqrt[3]{{\frac{a_5}{a_2}}}\)).

Geometric Sequence Formula

The geometric sequence formula is a vital tool for figuring out any term within a geometric sequence. It is defined as \(a_n = a_1 \cdot r^{n-1}\), where \(a_n\) is the nth term we wish to find, \(a_1\) is the first term in the sequence, \(r\) is the common ratio, and \(n\) is the term's position number, starting from 1 for the first term. This formula succinctly encapsulates the essence of the geometric sequence, showing how each term builds upon the preceding one through multiplication by the common ratio.

To illustrate, if you needed to discover the 10th term of a sequence where you know the first term and the common ratio, you would simply plug these values into the formula with \(n = 10\). By doing so, you're compiling the effect of the common ratio applied repeatedly, nine times in this case, to the first term.

Cube Root in Geometric Series

In geometric sequences, there may be situations where you need to find the cube root when dealing with terms of the series that are three positions apart. As seen in the exercise, when given \(a_2\) and \(a_5\), we calculated the common ratio by taking the cube root of their quotient. This is because the cube root is the inverse operation of cubing a number, which in the context of a geometric sequence, corresponds to applying the common ratio three times.

The cube root function is denoted by \(\sqrt[3]{x}\), where \(x\) is the result of a number cubed. It is essential in solving problems in geometric sequences where intervals of three terms are involved, and by extension, it is also used in other scenarios where the root corresponds to the term interval in question.