Problem 35 - 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. \( a_1 = 4, r = \dfrac{1}{2}, n = 10 \)

Step-by-Step Solution

Answer

The \( n \)th term of our geometric sequence, \( a_10 \), is approximately 0.0078125 or more accurately \( 4*(1/2)^{9} \).

Step 1: Understand the given parameters

In this exercise we have the first term \( a_1 = 4 \), the common ratio \( r = 1/2 \), and we are asked to find the 10th term, so \( n = 10 \).

Step 2: Apply the formula to compute nth term

The formula for the nth term of a geometric sequence is given by \( a_n = a_1 * r^{(n-1)} \). For the given geometric sequence, replacing \( a_1 \), \( r \) and \( n \) with 4, \( 1/2 \) and 10 respectively, we get \( a_n = 4 * (1/2)^{10-1} \).

Step 3: Evaluate the expression

Now calculate the value by solving the exponent first according to BODMAS rule (Brackets of Division Multiplication Addition Subtraction). Here, \( (1/2)^{9} \) will be evaluated first. Then the result is multiplied by 4 to get the final 10th term.

Key Concepts

Common Ratio

In a geometric sequence, the "common ratio" is a fundamental concept that defines the relationship between consecutive terms. Understanding the common ratio helps you identify how the sequence grows or shrinks.

The common ratio is the factor by which you multiply one term in the sequence to get to the next term. For a sequence \( a_1, a_2, a_3, \.\.\., a_n \), the common ratio \( r \) is calculated using the formula: \( r = \frac{a_{2}}{a_{1}} \).

In our exercise, the given common ratio is \( r = \frac{1}{2} \). This indicates that each term is half the size of the preceding term. Whether the common ratio is greater or less than 1 determines whether the sequence is increasing or decreasing, respectively. For instance:

• If the common ratio is positive and greater than 1, the sequence grows larger.
• If the common ratio is between 0 and 1, as in this exercise, the sequence decreases.

Recognizing the common ratio in any geometric sequence is crucial because it provides the "pulse" of the sequence, defining its specific pattern and behavior over time.

Nth Term Formula

The nth term formula is a powerful tool in geometry sequences, allowing you to find any term in the sequence without listing all previous terms. The formula provides a straightforward way to calculate the value of a specific term, particularly useful in longer sequences.

The nth term of a geometric sequence is computed using the formula:\[ a_n = a_1 \times r^{(n-1)} \]where:

• \( a_n \) is the term you want to find,
• \( a_1 \) is the first term in the sequence,
• \( r \) is the common ratio,
• \( n \) is the term number.

This formula allows you to plug in the values you have to calculate any desired term directly. For example, in our problem, with \( a_1 = 4 \), \( r = \frac{1}{2} \), and \( n = 10 \, \) you can compute the 10th term (\( a_{10} \)) using this formula, yielding \[ a_{10} = 4 \times \left(\frac{1}{2}\right)^{9} \].

Mastering this formula grants you the ability to predict future terms in the sequence without needing to individually calculate each prior term, saving time and minimizing potential error.

Arithmetic Operations

Arithmetic operations play a significant role when solving problems involving geometric sequences. In mathematics, arithmetic operations are the basic processes of addition, subtraction, multiplication, and division. In the context of calculating terms in a geometric sequence, these operations are a part of simplifying the sequence's expressions.

In our exercise, once the formula for the nth term is established, arithmetic operations help in simplifying the expression. Following the BODMAS/BIDMAS rules is essential: this involves carrying out operations inside brackets first, then exponents (orders), followed by division and multiplication, and finally addition and subtraction. Specifically:

• Calculate the exponent \( (1/2)^9 \) as the first step.
• After obtaining the result of the exponent, multiply it by \( a_1 \), which is 4.

This understanding ensures that you correctly simplify expressions to find the desired term value in the sequence. Precision in these operations ensures you get the correct outcome, making problem-solving more efficient and accurate. Using a calculator can assist in dealing with larger powers and fractions, providing a quick way to handle longer calculations accurately.