Problem 38 - Precalculus with Limits

Question

In Exercises 35 - 44, write an expression for the \( n \)th term of the geometric sequence. Then find the indicated term. \( a_1 = 64, r = -\dfrac{1}{4}, n = 10 \)

Step-by-Step Solution

Answer

Thus, the 10th term in the geometric sequence is -1/4096.

Step 1: Identify the Given Values

We know that the first term, \( a_1 \) is 64, the common ratio, r is -1/4 and we are required to find the 10th term that is \( n = 10 \).

Step 2: Apply the Geometric Sequence Formula

The nth term of a geometric sequence can be found by the formula \( a_n = a_1 \cdot r^{(n-1)} \). Substituting the given values, we get \( a_{10} = 64 \cdot (-\dfrac{1}{4})^{(10-1)} \).

Step 3: Solve the Expression

Solving the above expression, we get \( a_{10} = 64 \cdot (-\dfrac{1}{4})^{9} = 64 \cdot -\dfrac{1}{262144} = -\dfrac{1}{4096} \).

Key Concepts

nth term

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. The nth term of a geometric sequence can be determined using the "nth term formula." This formula is expressed as \( a_n = a_1 \cdot r^{(n-1)} \). Here, \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
Understanding how to use this formula is crucial in identifying any specific term in the sequence. For example, if the first term is 64 and the common ratio is \(-\frac{1}{4}\), you can find any term, like the 10th term, by plugging these values into the formula.
In our case, calculating the 10th term involves substituting n = 10 into the formula: \( a_{10} = 64 \times \left( -\frac{1}{4} \right)^{9} \). This step-by-step method helps decode any term's position in the sequence.

common ratio

The "common ratio" in a geometric sequence is the factor by which we multiply each term to get the next term. It is a defining characteristic of a geometric sequence.
Notably, the common ratio can be calculated by dividing any term by its preceding term. For example, if your sequence starts at 64 and the next term is -16, the common ratio \( r \) would be \(-\frac{1}{4}\). This means each term is obtained by multiplying the previous one by \(-\frac{1}{4}\).
It’s important to recognize that the common ratio can be positive or negative, fraction or whole number. A negative common ratio makes the terms alternate in sign, while a positive common ratio keeps all terms the same in sign. Also, a common ratio of less than 1 in absolute value results in terms decreasing in size, as seen in this sequence.

sequence formula

The "sequence formula" for a geometric sequence is foundational in predicting any term's value within the sequence. This powerful formula is given by \( a_n = a_1 \cdot r^{(n-1)} \). It allows us to calculate the nth term directly without listing all previous terms.
The formula's strength lies in its capacity for precision, tailoring to sequences with defined starting values and common ratios, like \( a_1 = 64 \) and \( r = -\frac{1}{4} \). When you need the 10th term, simply substitute: \( n = 10 \), \( a_1 = 64 \), and \( r = -\frac{1}{4} \) into the formula to find \( a_{10} = 64 \times (-\frac{1}{4})^{9} \).
By leveraging this formula, plotting any particular point in your sequence becomes straightforward and reliable, regardless of complexity.

series

A "series" in the context of geometric sequences refers to the sum of terms up to a certain point. While the exercise here focuses on finding a specific term, understanding the concept of a series is equally important.
The geometric series' sum can be determined using the formula \( S_n = a_1 \cdot \frac{1-r^n}{1-r} \), where \( S_n \) is the sum of the first n terms, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
A key point to consider is that if the absolute value of \( r \) is less than 1, the terms tend to zero as \( n \) becomes large, which significantly affects the sum. This understanding broadens your mathematical toolkit, preparing you for in-depth exploration of sequences and series in varied contexts.