Problem 9 - Precalculus Mathematics for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=3, \quad r=5$$

Step-by-Step Solution

Answer

The fourth term is 375.

Step 1: Understanding the Formula

The \(n\)th term of a geometric sequence is calculated using the formula \(a_n = a \, r^{(n-1)}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Identify Given Values

We are given \(a = 3\) and \(r = 5\). We need to find the fourth term, so \(n = 4\).

Step 3: Substitute Values into Formula

Substitute the known values into the \(n\)th term formula: \(a_4 = 3 \times 5^{(4-1)}\).

Step 4: Calculate the Exponent

Calculate \(5^{(4-1)} = 5^3\). Since \(5^3 = 125\), substitute this back into the expression.

Step 5: Final Calculation

Now substitute \(125\) in place of \(5^3\): \(a_4 = 3 \times 125 = 375\).

Key Concepts

nth term formula

The nth term formula in a geometric sequence is a fundamental tool. It helps us calculate any term in the sequence without listing all preceding terms. The formula is given by: \[ a_n = a \times r^{(n-1)} \]Where:

• \(a_n\) is the nth term of the sequence
• \(a\) is the first term
• \(r\) is the common ratio
• \(n\) is the term number you want to find

When using this formula, first identify your given values: the first term \(a\) and the common ratio \(r\). Then, determine which term \(n\) you want to calculate. Substitute these values into the formula. You'll use your calculator or perform mental math to evaluate the exponent \(r^{(n-1)}\). Once you multiply the base \(a\) by this result, you have the nth term of the sequence. Remember to follow the order of operations carefully, especially when dealing with exponents.

common ratio

The common ratio is a key feature of a geometric sequence. It determines how each term relates to the one before it. This ratio is constant throughout the sequence and is found by dividing any term by the preceding term. For example, in our sequence:

• If the first term \(a\) is 3 and the common ratio \(r\) is 5, then each term is made by multiplying the preceding term by 5.

To check if a sequence is geometric, simply verify that this relationship holds true for the entire sequence. The common ratio can be a positive number, a negative number, or even a fraction.

• When the common ratio is greater than 1, the sequence increases.
• When the common ratio is between 0 and 1, the sequence decreases.
• If the ratio is negative, the sequence will alternate signs (positive to negative, and so forth).

Understanding the common ratio helps predict future terms and understand the sequence's growth or decay.

geometric progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence can represent a variety of real-world situations, from population growth to financial interest calculations.The formula for the nth term of a geometric progression, \(a_n = a \times r^{(n-1)}\), allows us to determine any term in the sequence. In our original problem, to find the fourth term, we used the progression:

• First term \(a = 3\)
• Common ratio \(r = 5\)
• Fourth term \(a_4 = 3 \times 5^{3} = 375\)

A geometric progression can be visualized easily: each step either increases or decreases by the same factor due to the common ratio. Such sequences can showcase exponential growth or decay. Pay attention to the common ratio's sign and value because they deeply influence the behavior of the progression. For instance, a progression with a large common ratio grows quickly, while a small one grows slowly or even diminishes.